Motion Planning for Active Acceleration ... - Semantic Scholar
Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
Motion Planning for Active Acceleration Compensation Michael W. Decker [email protected]
Anh X. Dang [email protected]
Imme Ebert-Uphoff ∗ [email protected]
Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA, 30332-0405, USA Abstract The goal of this research is to enhance the capabilities of transport vehicles so that they can carry delicate objects of various shapes and sizes without requiring extensive packaging to protect them. This will be achieved, as first proposed by Graf and Dillmann (1997), by mounting a robotic device on top of the vehicle whose motion compensates for any forces or torques that act on the objects as a result of the vehicle’s motion, including disturbances caused by uneven terrain. This approach is called “Active Acceleration Compensation”. Several different approaches for the motion planning are implemented and compared. One degreeof-freedom motion planning algorithms include: (1) an algorithm based on optimal control theory (OCA), (2) an algorithm based on global optimization schemes (GMA), (3) a more flexible local optimization scheme based on feedback algorithms (FMPA). Finally, a three degrees-of-freedom motion planning algorithm, based on a combination of FMPA and a so called Pendulum Algorithm (3DOF-FMPA), is presented. Simulation results show that active acceleration compensation using these methods has the potential to significantly improve the performance of the system, as compared to (a) using no actuation at all; or (b) using the classic wash-out filter for motion planning.
1
Introduction
Automated Guided Vehicles (AGVs) and Mobile Robots have proven to be highly effective for the transport of material in many factory settings. However, this method of transportation is limited to objects that are not very delicate. Delicate objects are either carried manually or require a large amount of protective (and cumbersome) packaging in order to prevent damage during transport on a mobile robot or AGV. ∗ Please
address correspondence to this author.
0-7803-6475-9/01/$10.00© 2001 IEEE
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Classic examples of such delicate objects are hard disk platters and silicon wafers. At a cost of $500,000 per wafer, scratches must be prevented under any circumstances. Furthermore, pressed ceramic or metal parts that have not yet been sintered are very fragile and must be handled carefully as they are transported to the dryer and sintering furnace (examples are spark plugs, large electrical insulators like those used on power poles and dinnerware). Other applications include the transport of bulks of glassware and the delivery of chemicals or bio-medical samples that may not be subjected to sudden acceleration. The goal of this research is to enhance the capabilities of transport vehicles so that they can carry delicate objects of various shapes and sizes without requiring extensive packaging to protect them. This would increase flexibility and reduce turn-around time. To achieve this a robotic mechanism, namely a parallel platform manipulator (PPM), is to be mounted on top of the vehicle as shown in Figure 1. Objects to be transported would be placed on top of the PPM. The PPM is capable of moving in six degrees of freedom to compensate for any accelerations that would act on the objects as a result of the vehicle’s motion (including terrain disturbances). This type of compensation is called “Active Acceleration Compensation” (AAC) and was first introduced by Graf and Dillmann [1] in 1997. In practice, it is not expected that it is possible to completely eliminate all acceleration effects, due in part to sensor delays and errors and limited bandwidth of the robotic compensation device. It is rather anticipated that the acceleration can be reduced to a safe level sufficient to extend the use of transport vehicles to many novel manufacturing applications.
1.1
Scope and Organization of this Article
The ultimate goal is to compensate for motion in full six degrees-of-freedom. However, as the motion planning for acceleration compensation is highly complex and not yet well explored, this article considers
Tray or pallet will be mounted on top
Objects to be transported ma
Acceleration sensors and gyroscope will be mounted here
Tray or pallet mg
total force
a (acceleration of robot)
Figure 1: Classic Gough-Stewart Platform mounted on Mobile Robot compensation only for motion in a vertical plane, i.e. only motion in three degrees-of-freedom, (y, z, θ), is compensated for. Acceleration compensation is investigated (i) using a compensation mechanism with only one degree-of-freedom (tilting of the tray) and (ii) using a mechanism with three degrees-of-freedom (translation and rotation in a vertical plane). To the best of the authors’ knowledge, the only research reported on active acceleration compensation is by Graf and Dillmann [1, 2, 3, 4, 5]. Those authors propose to use the washout filter that is known from flight simulators for the motion planning. This article presents other algorithms for the motion planning that we believe are more efficient for this purpose. The remainder of this article presents: (1) Further background on active acceleration compensation (Subsections 1.2,1.3); (2) A dynamic model of the forces and moments acting on the object to be transported (Section 2); (3) A performance measure that quantifies the objectives of active acceleration compensation (Section 3); (4) Motion planning schemes for the robotic device that optimize the performance measure and thus minimize the effects of forces and moments acting on the objects (Section 4); (5) Test scenarios for the motion of the mobile robot (Section 5); (6) Numerical results and a comparison of the efficiency of the different algorithms (Section 6).
1.2
Basic Techniques for AAC
The uneven terrain, the slope of the terrain, or active motion of the robot may make it difficult, or impossible to safely transport fragile objects on a vehicle. To actively compensate for the disturbances, the parallel manipulator must keep the forces and moments within certain bounds to prevent any undesired motion of the transported object. As discussed by Graf and Dillmann [1], there are
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Vehicle accelerating to the right Figure 2: Technique A – Acceleration Compensation by G-tilt by a Robot (left) and a Waiter (right) two strategies used for this purpose: (A) Compensation by Tilting (G-tilt) and (B) Compensation by CounterAcceleration. Technique A: Compensation by Tilting The tray/pallet on which the objects are placed presents a physical constraint. Therefore forces that act downwards in a direction normal to the tray cannot do any work, i.e. they will result in no motion. To compensate for a linear acceleration of long duration the upper platform is thus tilted, so that the tray is normal to the total force acting on it, as shown in Figure 2. As a result the magnitude of the total acting force may vary, but its direction is always normal to the tray, so that the objects placed on it do not move. This involves only the three rotational degrees of freedom of the parallel manipulator and is best compared to the wrist motion of a waiter when balancing a tray, as shown on the right of Figure 2. This technique is suitable in particular to compensate for accelerations of long duration, such as caused by the acceleration of a vehicle (or mobile robot) and any slope of the terrain. It is anticipated that accelerations of this type can be compensated for fairly well. Technique B: Compensation by CounterAcceleration Sudden changes in the acceleration of the mobile robot can be compensated for by accelerating the upper platform of the parallel manipulator in the opposite direction so that, ideally, no acceleration reaches the objects on the tray. Due to the limited range of motion of the parallel manipulator, compensation of this type can only be generated for a short duration. This technique could be effective for high-frequency signals with low amplitude.
Object
M x, inertia Q
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Figure 4: Free Body Diagram of the Object
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Figure 3: Kinematics
1.3
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Kinematics
The position vector, 1 r1Q , to the center of mass of the object with respect to the fixed reference frame, {F1 }, is given by
Comparison to Flight Simulators
Acceleration compensation is closely related to the techniques used for the motion bases of flight simulators in the following sense: In both cases, a parallel manipulator is used to manipulate forces and moments. Flight simulators seek to create the illusion of forces and moments acting on the pilot (to simulate the airplane’s linear acceleration or air disturbances). Similarly, acceleration compensation seeks to eliminate forces and moments acting on objects. The most common algorithm for the motion planning of flight simulators is known as wash-out filter, see Reid and Nahon [6, 7, 8] and Baarspul [9]. The wash-out filter implements the equivalent of the two techniques described in Subsection 1.2 for flight simulators. Graf and Dillmann [1] first used the washout filter for active acceleration compensation. Washout filters have also been implemented for this research and their performance is compared to the performance of the other motion planning algorithms in Section 6.
2
N
Dynamic Model of Object on PPM
In order to minimize the acceleration effects on the object to be transported, a model of the forces and moments acting on the object must first be derived. Therefore the kinematics and kinetics of the system are discussed in this section. We assume the motion of the mobile robot to be in the vertical plane. Therefore both, the mobile robot and the parallel manipulator, have 3 degrees of freedom which can be described by the generalized coordinates (y1 , z1 , θ1 ) and (y2 , z2 , θ2 ), respectively (see Figure 3).
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1 r1Q
=
1 r12
+ 1 r23 + 1 r3Q ,
(1)
The leading subscript denotes the reference frame in which the quantities are expressed. The orientation of the object with respect to the fixed reference frame, {F1 }, is given by the expression (θ1 + θ2 ). The linear acceleration 1 aQ of the object Q with respect to the reference frame, {F1 }, is obtained by differentiating the postion vector twice with respect to time. The resulting expressions are rather lengthy and are omitted here, but can be found in [10].
2.2
Kinetics
The forces and moments acting on the object are investigated in this section. To derive a criterion for stability (Section 3), it is assumed that the object does not move relative to the top of the parallel manipulator. (When motion of the object relative to the top would occur can be detected nevertheless by looking at the accelerations acting on the object.) A free-body diagram of the object is shown in Figure 4. The external forces and moments acting on the object are gravity G, normal force N , friction force F and reaction moment Mreaction . The inertial forces Fy, inertia and Fz, inertia and also the inertial moment Mx, inertia oppose any angular or translational acceleration of the object according to Newton-Euler’s law. Newton-Euler’s method leads to three equations for the forces and moments acting on the object on the tray. For simplicity, the components of the acceleration vector 3 aQ along the axes 3 ey and 3 ez (see Figure 4) are denoted as ay and az . In order to be independent of the mass of the object to be transported, it is beneficial
to derive all equations in terms of accelerations and specific forces, i.e. force over mass: F = ay + g sin(θ1 + θ2 ) m
(2)
N = az + g cos(θ1 + θ2 ) m
(3)
Mreaction + F ∆z = Ixx (θ¨1 + θ¨2 )
(4)
Equations (2) and (3) can be expanded by substituting the kinematic expression for the accelerations. The resulting expressions are very lengthy (see [10]) and are omitted here. The resulting expressions are used in the performance function derived in the following section.
3
Performance Cost Function L
This section seeks to establish a performance measure for the stability of the object. In defining stability, we are concerned with the object’s tendency to slide, rotate, or lift off relative to the PPM. Thus the word stable is used to indicate that the object on top of the parallel manipulator remains at its initial position with respect to the tray. The critical quantities, which act as the main indicators for instability, are the following: N m,
where N is the 1. The specific normal force normal force that the tray exerts on the object and m is the object’s mass. N m should be positive and large to keep the object from lifting. F , with F being the required friction 2. The quantity N force tangential to the bottom surface of the object F should be small to keep the object and the tray. N from tending to slide.
3. The moment of reaction Mreaction , which is the reaction moment about the center of mass of the object due to the object’s inertia. Mreaction should be small to keep the object from tipping. To assess the performance of the motion planning algorithm at each moment in time, it is helpful to express conditions of the system by means of a performance function. In this research, a performance function is chosen to represent the stability of the object and the readiness of the PPM by means of a single numerical value. The motion planning of the parallel manipulator can then be formulated as a minimization problem with respect to the control variables: y¨2 , z¨2 and θ¨2 .
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The explicit expression for the performance function L is chosen as follows: 2 F N L = c1 − c2 (5) N m h N N i − − c3 atan c4 m limit i h m − c5 atan c6 nmin − nmin, limit h F i + c7 atan c8 − µs h N i + c9 atan c10 θ2 − θ2, limit h i + c11 atan c12 θ˙2 − θ˙2, limit L is a sum of weighted functions, each performing a specific task that is related to either the stability, workspace or actuation limits of the system. The performance function is formulated such that it yields minimal values when the friction force is small, the normal force is large, and the system is operating well within workspace and actuation limits. If one or more of the constraints is exceeded or close to being exceeded, the performance function yields extremely large values. For a detailed motivation and derivation of this particular performance function, see [10].
4
Motion Planning Algorithms
This section discusses different approaches that were implemented for the motion planning for active acceleration compensation. The input for the motion planning algorithms is always the trajectory of the mobile robot (denoted as I(t) in the following) and the output is the optimized linear and/or rotary motion of the parallel manipulator over time (denoted as O(t)). In the following, the motion planning algorithms are categorized by their computational characteristics and by the number of actuated degrees-of freedom (DOF).
4.1
One-DOF Approaches
This section discusses different approaches to implement the compensation by only tilting the PPM. The input, I(t), is the trajectory of the mobile robot over time, in terms of y1 (t), z1 (t), θ1 (t). The output, O(t), is the tilting angle over time, θ2 (t). 4.1.1
One-DOF Offline Approaches
For all offline approaches the trajectory of the mobile robot must be known ahead of time for the entire considered time period [t0 , tf inal ].
(1) Optimal Control Algorithm (OCA) Description: The solution of the Optimal Control Algorithm minimizes the integral of the performance Rtf cost function, L(t)dt. t0
The advantage of this approach is that the well established framework of optimal control can be used. However, disadvantages include the fact that the integral of the performance function is minimized by this approach, while it is really the maximum of the performance function in [t0 , tf ] that we seek to minimize. Furthermore, convergence is only guaranteed when good initial guesses for the algorithm’s start values are available. As will be seen from the numerical results, the OCA yields sub-optimal performance (Section 6). (2) Global Minimization Algorithm (GMA) Description: The output function, O(t), is represented by a finite sum of trigonometric or polynomial functions. The coefficients of the representation are then optimized such that the maximum of the performance function over the time period is minimized. An advantage of this method is that it seeks to minimize in fact the maximal value of the performance function L within [t0 , tf ]. However, the search space of functions O(t) is restricted to functions represented by a finite, small sum of trigonometric or polynomial functions, which does not always include a close approximation of the optimal solution. As one would anticipate, an approximation using sine functions yields much better results than polynomials. As will be seen from the numerical results, however, even the GMA based on sine functions yields only sub-optimal performance (Section 6). 4.1.2
One-DOF Online Approach: Feedback Motion Planning Algorithm (FMPA)
In contrast to the offline approaches, the FMPA is an online approach. This implies that to calculate the output motion, O(t∗ ), at time t∗ , the input motion, I(t), only has to be known up to time t∗ . Online approaches are essential for any practical application where sensor signals are used to determine an unknown trajectory of the mobile robot. The FMPA is a local optimization scheme that determines the optimal angular acceleration of the platform, θ¨2 , at discrete time steps, ti , in order to optimize performance function L. The FMPA obtains the word “Feedback” in its title from the fact that in addition to the input motion, I(ti ), the output motion at previous time steps, O(ti−1 ) is fed back into the minimization routine. This is required to assure that the acceleration caused by the parallel manipulator
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y1
LP Sensor Data from Mobile Robot
z1 ..
+
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Actuation Signals for Parallel Manipulator
θ2
+ +
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Figure 5: Algorithm
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Pendulum Algorithm
(a) ..
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Structure of 3-DOF Motion Planning
itself is taken into account when calculating forces and moments acting on the object. In order to yield good results, the FMPA includes interpolation algorithms that seek to predict the input motion at the next time step, ti+1 , based on prior and current input motion, to decide what action to take at time ti . The local minimization routine utilizes the golden section search and parabolic interpolation algorithms. The FMPA is the only scheme (of the three 1-DOF schemes considered) that can be performed online, and it turns out that in general it even yields the best results (see Section 6).
4.2
Three-DOF Online Approach (3DOFFMPA)
This section discusses the extension of the FMPA algorithm to optimize not only the tilting of the PPM, but also its translation in the vertical plane for optimal acceleration compensation. As in the previous case, the input, I(t), is the trajectory of the mobile robot over time, in terms of y1 (t), z1 (t), θ1 (t). The output, O(t), is now the translation and rotation of the parallel manipulator over time, y2 (t), z2 (t), θ2 (t). The structure of the 3-DOF Motion Planning Algorithm is shown in Figure 5. It has two main components: (a) The block in the lower half uses the FMPA to calculate the optimal angular acceleration, θ¨2 , of the PPM as in the one-DOF algorithm. (b) The block in the upper half employs a so-called Pendulum Algorithm to calculate the optimal linear acceleration, y¨2 , z¨2 , of the PPM. The remaining components of the diagram (LP,MRP,HP) represent low-pass, mid-rangepass and high-pass filters. The components of the trajectory of the mobile robots are filtered into low, mid-range, and high frequencies. For this application, all high frequency signals are ignored for the following reason. In practice, inherent delays in the system will not allow the parallel
.. z1 1ez
robot is required. Many different scenarios were tested and following two typical scenarios were selected for this article: the Horizontal and Wave scenarios.
.. y1
Frame {1} 1ey
O1
5.1 α
rod
Horizontal Scenario
The Horizontal Scenario simulates a smooth halfsinusoidal acceleration in the horizontal driving direction of the mobile robot.
l Gravity
F
30 20 10 0
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Figure 6: Object on Pendulum manipulator to respond fast enough to compensate for any acceleration input. The limit of 3 Hz is currently used as the limit of what the parallel manipulator can respond to. The remaining low and mid-range signals are fed into the proper channels of the 3DOF-FMPA algorithm as shown in Figure 5. In particular, to calculate the optimal angular acceleration, θ¨2 , of the parallel manipulator, the low frequency component is fed into the FMPA algorithm, while the mid-range frequency signal is simply negated to “match” the mobile robot’s motion. The pendulum algorithm emulates the response of a virtual pendulum (shown in Figure 6) when subjected to linear accelerations applied at its pivot point. The object is attached to the virtual pendulum as shown in Figure 6. This pendulum approach is motivated by the fact that the free swinging motion of a pendulum automatically minimizes the lateral force acting on the object. For this application, the linear acceleration, y¨1 , z¨1 , of the mobile robot is mapped to the linear acceleration of the pendulum’s pivot point. The resulting linear motion of the pendulum’s tray is mapped to the linear acceleration of the PPM. The pendulum algorithm is suitable for input trajectories with low to midrange frequency y¨1 and low frequency z¨1 linear acceleration. (This is because the pendulum motion to any relatively high frequency z¨1 acceleration input will not result in beneficial motion in terms of acceleration compensation.) Therefore, the midrange frequency potion of the z¨1 signal is filtered and negated, analogous to the θ¨2 signal. For details, see [10].
5
Motion Scenarios for Mobile Robot
To test the effectiveness of the motion planning algorithm some sample input motion of the mobile
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Figure 7: Horizontal Scenario: Mobile Robot only moves in y1 -direction As illustrated in Figure 7, the robot moves horizontally with a maximum acceleration of 4 sm2 until it reaches a constant velocity. No changes in orientation occur, i.e. angle θ1 (t) = 0 . This scenario simulates the realistic case when the mobile robot accelerates from rest to a certain constant velocity. The deceleration scenario would have symmetric effects on the actuation of the parallel platform and is therefore not investigated explicitly.
5.2
Wave Scenario
The Wave Scenario provides a tool to test the response of the motion planning schemes for the parallel manipulator to a highly dynamic input. The upper graph in Figure 8 shows the motion of the mobile robot in the y1 − z1 plane: the mobile robot moves on a sinusoidal curve followed by a horizontal plateau. The remaining graphs in Figure 8 show the separate components, y1 , z1 , θ1 of the trajectory of the mobile robot over time. The mobile robot is simulated to move along this path with its orientation θ1 always tangential to it. The constant velocity v along the path, the frequency ω of the sinusoidal part, and the factor f controlling the amplitude can all be specified by the user.
6
Simulation Results
To assess the performance of different motion planning algorithms, the values of the residual forces (normal and friction) and reaction moments of the object on the PPM are compared to those without any actuation. The residual forces and moments are calculated from Equations (2), (3), (4) with the outputs of the motion planning algorithms. Dividing the specific normal
Plot of the Trajectory in YZ Plane
Horizontal Input Trajectory Results
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scenario and the 3-DOF algorithms for Wave scenario are shown here. 0
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Figure 9: Comparison of 1-DOF Algorithms Results
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Figure 8: Wave Scenario: Trajectory of Mobile Robot F force, N m , by the specific friction force, N , provides the required friction coefficient, µ, to keep the object from F > 0). This quantity characterizes sliding (assumed N the stability of the object from sliding. Therefore, the simulation results of µ, instead of F and N, are presented here. Due to space limitation, results for the reaction moment can be found in [10]. However, in general, it can be said that the trend for the reaction moment, Mreaction , follows that of µ. Therefore, by looking at µ, one can induce the trend in Mreaction . This is simply a rule of thumb induced from the data set. In addition to comparing the relative values of the residual forces and moments to assess performance of the different motion algorithms, these values can be compared with threshold values that would make the object on the PPM unstable. In practice, these threshold values depend on the dimensions and mass of the object, as well as the parameters of the surface on which it resides. Due to these uncertainties in object dimensions and surface parameters, only the stability for one set of fixed threshold values were explored. The results of these findings can be found in [10]. Simulation of the motion planning algorithms was done in the Matlab environment, with the Simulink Toolbox. Due to space limitations, only the results of the 1-DOF motion planning algorithms for Horizontal
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One-DOF Motion Planning Results
The results of the one-DOF motion planning algorithms are shown in Figure 9. These results are for the Horizontal scenario and only the angle of the PPM is actuated (no linear actuation). There are four sets of data in this figure. The dashed line indicates the required friction coefficient to keep the object stable on the PPM without any actuation (compensation turned off). The other lines show the performance of the FMPA, the GMA based on sine functions and the OCA algorithm, in reducing the required friction coefficient. In general, the FMPA online scheme has the lowest value of µ over the long periods of time. As is apparent from the figure, the motion planning scheme drastically improves the performance of the system, as measured by the level of stability of the object.
6.2
Three-DOF Motion Planning Results
A comparison of the required friction coefficient for the 3DOF-FMPA, washout filter and the unactuated case are compared in Figure 10. Notice the drastic improvement that the 3DOF-FMPA provides over the unactuated and washout filter schemes. The friction coefficient is close to zero until the mobile robot reaches the 10 second mark and has to transition from an uphill climb to a level terrain. This transition requires some level of static friction to help the object remain stable. This level is still significantly below the unactuated and washout filter case. For other scenarios the washout filter performed better than in the example shown in Figure 10. Furthermore, the performance of the washout filter depends heavily on the selection of the filter parameters and may leave room for improvement. Nevertheless, for the parameters chosen here, its results were always significantly worse than that of the 3DOF-FMPA.
Wave Input Trajectory Results
(Studienstiftung), both awarded to Michael Decker. In addition this research was supported by a doctoral fellowship of the Engineering Research Program of the Office of Basic Energy Sciences at the Department of Energy, awarded to Anh Dang.
0.25 W/O Actuation Washout Filter 3DOFFMPA
Required Friction Coeff
0.2
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References 0.1
[1] R. Graf and R. Dillmann, “Active acceleration compensation using a Stewart-platform on a mobile robot,” in Proceedings of the 2nd Euromicro Workshop on Advanced Mobile Robots, EUROBOT 97, (Brescia, Italy), pp. 59 – 64, 1997.
3DOFFMPA 0.05
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Figure 10: Comparison of 3-DOF Algorithm and Washout Filter Results
7
Conclusions and Future Work
Several strategies were proposed for active acceleration compensation to facilitate the transport of delicate objects. A performance measure for the system was developed on which all proposed motion planning algorithms were based. When considering the motion planning for the tilting of the parallel manipulator, both off-line (OCA & GMA) and online (FMPA) motion planning schemes were assessed. For the full three DOF actuating scheme, the 3DOF-FMPA was developed and compared with the motion planning algorithm called the washout filter. For the one-DOF Horizontal scenario case, simulation results showed that the actuation schemes of both the off-line and on-line motion planning algorithms improved the stability of an object on a PPM, and that the online motion planning scheme gave better results than the off-line schemes. Even better system performance was achieved by the full three-DOF actuation scheme, which out-performed the washout-filter motion planning scheme. Encouraged by the improvements that can be obtained even with a 1-DOF device, a prototype of a 1-DOF acceleration compensation device is currently being constructed at Georgia Tech. Future work includes the integration of sensors, experiments with the 1-DOF prototype and the construction of a 3-DOF prototype.
8
Acknowledgements
This work was supported by fellowships of the German Academic Exchange Service (DAAD) and the German National Merit Scholarship Foundation
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[2] R. Graf and R. Dillmann, “Aktive beschleunigungskompensation mittels einer Stewart-Plattform auf einem mobilen roboter (in German),” in Proceedings of Autonome Mobile Systeme - AMS 97, (Stuttgart, Germany), p. unknown, 1997. Available at http://wwwipr.ira.uka.de/publications/papers.html. [3] R. Graf and R. Dillmann, “Acceleration compensation using a Stewart-platform on a mobile robot,” in Proceedings of the 3rd Euromicro Workshop on Advanced Mobile Robots, EUROBOT 99, (Zuerich, Switzerland), pp. 17 – 24, Sept 6-8 1999. [4] R. Graf and R. Dillmann, “Die Stewart-Plattform als dynamisches lastaufnahmesystem eines mobilen roboters (in German),” in Proceedings of Autonome Mobile Systeme - AMS 99, (Munich, Germany), pp. 150 – 159, 1999. [5] R. Graf and R. Dillmann, “Ein flexibles lastaufnahmesystem - die Stewart-Plattform (in German),” in Robotik 2000, p. unknown, 2000. Available at http://wwwipr.ira.uka.de/publications/papers.html. [6] M. Nahon and L. Reid, “Simulator motion-drive algorithms. a designer’s perspective,” Journal of Guidance, Control, and Dynamics, vol. 13, pp. 356– 362, March–April 1990. [7] L. Reid and M. Nahon, “Flight simulation motionbase drive algorithms. part 1: Developing and testing the equations.,” Tech. Rep. UTIAS 296PT1, Toronto University, Inst. for Aerospace Studies, December 1985. [8] L. Reid and M. Nahon, “Flight simulation motion-base drive algorithms. part 2: Selecting the system parameters.,” Tech. Rep. UTIAS 307, Toronto University, May 1986. [9] M. Baarspul, “Lecture notes on flight simulation techniques,” Tech. Rep. LR-596, Technische Hogeschool Delft (Netherlands). Faculty of Aerospace Engineering, 1989. [10] M. Decker, “Active acceleration compensation for transport of delicate objects,” Master’s thesis, Georgia Institute of Technology, Atlanta, Georgia, August 2000.
Motion Planning for Active Acceleration Compensation Michael W. Decker [email protected]
Anh X. Dang [email protected]
Imme Ebert-Uphoff ∗ [email protected]
Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA, 30332-0405, USA Abstract The goal of this research is to enhance the capabilities of transport vehicles so that they can carry delicate objects of various shapes and sizes without requiring extensive packaging to protect them. This will be achieved, as first proposed by Graf and Dillmann (1997), by mounting a robotic device on top of the vehicle whose motion compensates for any forces or torques that act on the objects as a result of the vehicle’s motion, including disturbances caused by uneven terrain. This approach is called “Active Acceleration Compensation”. Several different approaches for the motion planning are implemented and compared. One degreeof-freedom motion planning algorithms include: (1) an algorithm based on optimal control theory (OCA), (2) an algorithm based on global optimization schemes (GMA), (3) a more flexible local optimization scheme based on feedback algorithms (FMPA). Finally, a three degrees-of-freedom motion planning algorithm, based on a combination of FMPA and a so called Pendulum Algorithm (3DOF-FMPA), is presented. Simulation results show that active acceleration compensation using these methods has the potential to significantly improve the performance of the system, as compared to (a) using no actuation at all; or (b) using the classic wash-out filter for motion planning.
1
Introduction
Automated Guided Vehicles (AGVs) and Mobile Robots have proven to be highly effective for the transport of material in many factory settings. However, this method of transportation is limited to objects that are not very delicate. Delicate objects are either carried manually or require a large amount of protective (and cumbersome) packaging in order to prevent damage during transport on a mobile robot or AGV. ∗ Please
address correspondence to this author.
0-7803-6475-9/01/$10.00© 2001 IEEE
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Classic examples of such delicate objects are hard disk platters and silicon wafers. At a cost of $500,000 per wafer, scratches must be prevented under any circumstances. Furthermore, pressed ceramic or metal parts that have not yet been sintered are very fragile and must be handled carefully as they are transported to the dryer and sintering furnace (examples are spark plugs, large electrical insulators like those used on power poles and dinnerware). Other applications include the transport of bulks of glassware and the delivery of chemicals or bio-medical samples that may not be subjected to sudden acceleration. The goal of this research is to enhance the capabilities of transport vehicles so that they can carry delicate objects of various shapes and sizes without requiring extensive packaging to protect them. This would increase flexibility and reduce turn-around time. To achieve this a robotic mechanism, namely a parallel platform manipulator (PPM), is to be mounted on top of the vehicle as shown in Figure 1. Objects to be transported would be placed on top of the PPM. The PPM is capable of moving in six degrees of freedom to compensate for any accelerations that would act on the objects as a result of the vehicle’s motion (including terrain disturbances). This type of compensation is called “Active Acceleration Compensation” (AAC) and was first introduced by Graf and Dillmann [1] in 1997. In practice, it is not expected that it is possible to completely eliminate all acceleration effects, due in part to sensor delays and errors and limited bandwidth of the robotic compensation device. It is rather anticipated that the acceleration can be reduced to a safe level sufficient to extend the use of transport vehicles to many novel manufacturing applications.
1.1
Scope and Organization of this Article
The ultimate goal is to compensate for motion in full six degrees-of-freedom. However, as the motion planning for acceleration compensation is highly complex and not yet well explored, this article considers
Tray or pallet will be mounted on top
Objects to be transported ma
Acceleration sensors and gyroscope will be mounted here
Tray or pallet mg
total force
a (acceleration of robot)
Figure 1: Classic Gough-Stewart Platform mounted on Mobile Robot compensation only for motion in a vertical plane, i.e. only motion in three degrees-of-freedom, (y, z, θ), is compensated for. Acceleration compensation is investigated (i) using a compensation mechanism with only one degree-of-freedom (tilting of the tray) and (ii) using a mechanism with three degrees-of-freedom (translation and rotation in a vertical plane). To the best of the authors’ knowledge, the only research reported on active acceleration compensation is by Graf and Dillmann [1, 2, 3, 4, 5]. Those authors propose to use the washout filter that is known from flight simulators for the motion planning. This article presents other algorithms for the motion planning that we believe are more efficient for this purpose. The remainder of this article presents: (1) Further background on active acceleration compensation (Subsections 1.2,1.3); (2) A dynamic model of the forces and moments acting on the object to be transported (Section 2); (3) A performance measure that quantifies the objectives of active acceleration compensation (Section 3); (4) Motion planning schemes for the robotic device that optimize the performance measure and thus minimize the effects of forces and moments acting on the objects (Section 4); (5) Test scenarios for the motion of the mobile robot (Section 5); (6) Numerical results and a comparison of the efficiency of the different algorithms (Section 6).
1.2
Basic Techniques for AAC
The uneven terrain, the slope of the terrain, or active motion of the robot may make it difficult, or impossible to safely transport fragile objects on a vehicle. To actively compensate for the disturbances, the parallel manipulator must keep the forces and moments within certain bounds to prevent any undesired motion of the transported object. As discussed by Graf and Dillmann [1], there are
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Vehicle accelerating to the right Figure 2: Technique A – Acceleration Compensation by G-tilt by a Robot (left) and a Waiter (right) two strategies used for this purpose: (A) Compensation by Tilting (G-tilt) and (B) Compensation by CounterAcceleration. Technique A: Compensation by Tilting The tray/pallet on which the objects are placed presents a physical constraint. Therefore forces that act downwards in a direction normal to the tray cannot do any work, i.e. they will result in no motion. To compensate for a linear acceleration of long duration the upper platform is thus tilted, so that the tray is normal to the total force acting on it, as shown in Figure 2. As a result the magnitude of the total acting force may vary, but its direction is always normal to the tray, so that the objects placed on it do not move. This involves only the three rotational degrees of freedom of the parallel manipulator and is best compared to the wrist motion of a waiter when balancing a tray, as shown on the right of Figure 2. This technique is suitable in particular to compensate for accelerations of long duration, such as caused by the acceleration of a vehicle (or mobile robot) and any slope of the terrain. It is anticipated that accelerations of this type can be compensated for fairly well. Technique B: Compensation by CounterAcceleration Sudden changes in the acceleration of the mobile robot can be compensated for by accelerating the upper platform of the parallel manipulator in the opposite direction so that, ideally, no acceleration reaches the objects on the tray. Due to the limited range of motion of the parallel manipulator, compensation of this type can only be generated for a short duration. This technique could be effective for high-frequency signals with low amplitude.
Object
M x, inertia Q
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Figure 4: Free Body Diagram of the Object
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Figure 3: Kinematics
1.3
2.1
Kinematics
The position vector, 1 r1Q , to the center of mass of the object with respect to the fixed reference frame, {F1 }, is given by
Comparison to Flight Simulators
Acceleration compensation is closely related to the techniques used for the motion bases of flight simulators in the following sense: In both cases, a parallel manipulator is used to manipulate forces and moments. Flight simulators seek to create the illusion of forces and moments acting on the pilot (to simulate the airplane’s linear acceleration or air disturbances). Similarly, acceleration compensation seeks to eliminate forces and moments acting on objects. The most common algorithm for the motion planning of flight simulators is known as wash-out filter, see Reid and Nahon [6, 7, 8] and Baarspul [9]. The wash-out filter implements the equivalent of the two techniques described in Subsection 1.2 for flight simulators. Graf and Dillmann [1] first used the washout filter for active acceleration compensation. Washout filters have also been implemented for this research and their performance is compared to the performance of the other motion planning algorithms in Section 6.
2
N
Dynamic Model of Object on PPM
In order to minimize the acceleration effects on the object to be transported, a model of the forces and moments acting on the object must first be derived. Therefore the kinematics and kinetics of the system are discussed in this section. We assume the motion of the mobile robot to be in the vertical plane. Therefore both, the mobile robot and the parallel manipulator, have 3 degrees of freedom which can be described by the generalized coordinates (y1 , z1 , θ1 ) and (y2 , z2 , θ2 ), respectively (see Figure 3).
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1 r1Q
=
1 r12
+ 1 r23 + 1 r3Q ,
(1)
The leading subscript denotes the reference frame in which the quantities are expressed. The orientation of the object with respect to the fixed reference frame, {F1 }, is given by the expression (θ1 + θ2 ). The linear acceleration 1 aQ of the object Q with respect to the reference frame, {F1 }, is obtained by differentiating the postion vector twice with respect to time. The resulting expressions are rather lengthy and are omitted here, but can be found in [10].
2.2
Kinetics
The forces and moments acting on the object are investigated in this section. To derive a criterion for stability (Section 3), it is assumed that the object does not move relative to the top of the parallel manipulator. (When motion of the object relative to the top would occur can be detected nevertheless by looking at the accelerations acting on the object.) A free-body diagram of the object is shown in Figure 4. The external forces and moments acting on the object are gravity G, normal force N , friction force F and reaction moment Mreaction . The inertial forces Fy, inertia and Fz, inertia and also the inertial moment Mx, inertia oppose any angular or translational acceleration of the object according to Newton-Euler’s law. Newton-Euler’s method leads to three equations for the forces and moments acting on the object on the tray. For simplicity, the components of the acceleration vector 3 aQ along the axes 3 ey and 3 ez (see Figure 4) are denoted as ay and az . In order to be independent of the mass of the object to be transported, it is beneficial
to derive all equations in terms of accelerations and specific forces, i.e. force over mass: F = ay + g sin(θ1 + θ2 ) m
(2)
N = az + g cos(θ1 + θ2 ) m
(3)
Mreaction + F ∆z = Ixx (θ¨1 + θ¨2 )
(4)
Equations (2) and (3) can be expanded by substituting the kinematic expression for the accelerations. The resulting expressions are very lengthy (see [10]) and are omitted here. The resulting expressions are used in the performance function derived in the following section.
3
Performance Cost Function L
This section seeks to establish a performance measure for the stability of the object. In defining stability, we are concerned with the object’s tendency to slide, rotate, or lift off relative to the PPM. Thus the word stable is used to indicate that the object on top of the parallel manipulator remains at its initial position with respect to the tray. The critical quantities, which act as the main indicators for instability, are the following: N m,
where N is the 1. The specific normal force normal force that the tray exerts on the object and m is the object’s mass. N m should be positive and large to keep the object from lifting. F , with F being the required friction 2. The quantity N force tangential to the bottom surface of the object F should be small to keep the object and the tray. N from tending to slide.
3. The moment of reaction Mreaction , which is the reaction moment about the center of mass of the object due to the object’s inertia. Mreaction should be small to keep the object from tipping. To assess the performance of the motion planning algorithm at each moment in time, it is helpful to express conditions of the system by means of a performance function. In this research, a performance function is chosen to represent the stability of the object and the readiness of the PPM by means of a single numerical value. The motion planning of the parallel manipulator can then be formulated as a minimization problem with respect to the control variables: y¨2 , z¨2 and θ¨2 .
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The explicit expression for the performance function L is chosen as follows: 2 F N L = c1 − c2 (5) N m h N N i − − c3 atan c4 m limit i h m − c5 atan c6 nmin − nmin, limit h F i + c7 atan c8 − µs h N i + c9 atan c10 θ2 − θ2, limit h i + c11 atan c12 θ˙2 − θ˙2, limit L is a sum of weighted functions, each performing a specific task that is related to either the stability, workspace or actuation limits of the system. The performance function is formulated such that it yields minimal values when the friction force is small, the normal force is large, and the system is operating well within workspace and actuation limits. If one or more of the constraints is exceeded or close to being exceeded, the performance function yields extremely large values. For a detailed motivation and derivation of this particular performance function, see [10].
4
Motion Planning Algorithms
This section discusses different approaches that were implemented for the motion planning for active acceleration compensation. The input for the motion planning algorithms is always the trajectory of the mobile robot (denoted as I(t) in the following) and the output is the optimized linear and/or rotary motion of the parallel manipulator over time (denoted as O(t)). In the following, the motion planning algorithms are categorized by their computational characteristics and by the number of actuated degrees-of freedom (DOF).
4.1
One-DOF Approaches
This section discusses different approaches to implement the compensation by only tilting the PPM. The input, I(t), is the trajectory of the mobile robot over time, in terms of y1 (t), z1 (t), θ1 (t). The output, O(t), is the tilting angle over time, θ2 (t). 4.1.1
One-DOF Offline Approaches
For all offline approaches the trajectory of the mobile robot must be known ahead of time for the entire considered time period [t0 , tf inal ].
(1) Optimal Control Algorithm (OCA) Description: The solution of the Optimal Control Algorithm minimizes the integral of the performance Rtf cost function, L(t)dt. t0
The advantage of this approach is that the well established framework of optimal control can be used. However, disadvantages include the fact that the integral of the performance function is minimized by this approach, while it is really the maximum of the performance function in [t0 , tf ] that we seek to minimize. Furthermore, convergence is only guaranteed when good initial guesses for the algorithm’s start values are available. As will be seen from the numerical results, the OCA yields sub-optimal performance (Section 6). (2) Global Minimization Algorithm (GMA) Description: The output function, O(t), is represented by a finite sum of trigonometric or polynomial functions. The coefficients of the representation are then optimized such that the maximum of the performance function over the time period is minimized. An advantage of this method is that it seeks to minimize in fact the maximal value of the performance function L within [t0 , tf ]. However, the search space of functions O(t) is restricted to functions represented by a finite, small sum of trigonometric or polynomial functions, which does not always include a close approximation of the optimal solution. As one would anticipate, an approximation using sine functions yields much better results than polynomials. As will be seen from the numerical results, however, even the GMA based on sine functions yields only sub-optimal performance (Section 6). 4.1.2
One-DOF Online Approach: Feedback Motion Planning Algorithm (FMPA)
In contrast to the offline approaches, the FMPA is an online approach. This implies that to calculate the output motion, O(t∗ ), at time t∗ , the input motion, I(t), only has to be known up to time t∗ . Online approaches are essential for any practical application where sensor signals are used to determine an unknown trajectory of the mobile robot. The FMPA is a local optimization scheme that determines the optimal angular acceleration of the platform, θ¨2 , at discrete time steps, ti , in order to optimize performance function L. The FMPA obtains the word “Feedback” in its title from the fact that in addition to the input motion, I(ti ), the output motion at previous time steps, O(ti−1 ) is fed back into the minimization routine. This is required to assure that the acceleration caused by the parallel manipulator
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..
..
y1
LP Sensor Data from Mobile Robot
z1 ..
+
..
∆θ2 '
Actuation Signals for Parallel Manipulator
θ2
+ +
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Figure 5: Algorithm
..
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z2
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(b)
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Pendulum Algorithm
(a) ..
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θ2 '
Structure of 3-DOF Motion Planning
itself is taken into account when calculating forces and moments acting on the object. In order to yield good results, the FMPA includes interpolation algorithms that seek to predict the input motion at the next time step, ti+1 , based on prior and current input motion, to decide what action to take at time ti . The local minimization routine utilizes the golden section search and parabolic interpolation algorithms. The FMPA is the only scheme (of the three 1-DOF schemes considered) that can be performed online, and it turns out that in general it even yields the best results (see Section 6).
4.2
Three-DOF Online Approach (3DOFFMPA)
This section discusses the extension of the FMPA algorithm to optimize not only the tilting of the PPM, but also its translation in the vertical plane for optimal acceleration compensation. As in the previous case, the input, I(t), is the trajectory of the mobile robot over time, in terms of y1 (t), z1 (t), θ1 (t). The output, O(t), is now the translation and rotation of the parallel manipulator over time, y2 (t), z2 (t), θ2 (t). The structure of the 3-DOF Motion Planning Algorithm is shown in Figure 5. It has two main components: (a) The block in the lower half uses the FMPA to calculate the optimal angular acceleration, θ¨2 , of the PPM as in the one-DOF algorithm. (b) The block in the upper half employs a so-called Pendulum Algorithm to calculate the optimal linear acceleration, y¨2 , z¨2 , of the PPM. The remaining components of the diagram (LP,MRP,HP) represent low-pass, mid-rangepass and high-pass filters. The components of the trajectory of the mobile robots are filtered into low, mid-range, and high frequencies. For this application, all high frequency signals are ignored for the following reason. In practice, inherent delays in the system will not allow the parallel
.. z1 1ez
robot is required. Many different scenarios were tested and following two typical scenarios were selected for this article: the Horizontal and Wave scenarios.
.. y1
Frame {1} 1ey
O1
5.1 α
rod
Horizontal Scenario
The Horizontal Scenario simulates a smooth halfsinusoidal acceleration in the horizontal driving direction of the mobile robot.
l Gravity
F
30 20 10 0
N
0
2
4
6
8
y1−ddot [ m/s2 ]
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Tray
y1−dot [ m/s ]
40
Object Q
6 4 2 0
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4 3 2 1 0
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Figure 6: Object on Pendulum manipulator to respond fast enough to compensate for any acceleration input. The limit of 3 Hz is currently used as the limit of what the parallel manipulator can respond to. The remaining low and mid-range signals are fed into the proper channels of the 3DOF-FMPA algorithm as shown in Figure 5. In particular, to calculate the optimal angular acceleration, θ¨2 , of the parallel manipulator, the low frequency component is fed into the FMPA algorithm, while the mid-range frequency signal is simply negated to “match” the mobile robot’s motion. The pendulum algorithm emulates the response of a virtual pendulum (shown in Figure 6) when subjected to linear accelerations applied at its pivot point. The object is attached to the virtual pendulum as shown in Figure 6. This pendulum approach is motivated by the fact that the free swinging motion of a pendulum automatically minimizes the lateral force acting on the object. For this application, the linear acceleration, y¨1 , z¨1 , of the mobile robot is mapped to the linear acceleration of the pendulum’s pivot point. The resulting linear motion of the pendulum’s tray is mapped to the linear acceleration of the PPM. The pendulum algorithm is suitable for input trajectories with low to midrange frequency y¨1 and low frequency z¨1 linear acceleration. (This is because the pendulum motion to any relatively high frequency z¨1 acceleration input will not result in beneficial motion in terms of acceleration compensation.) Therefore, the midrange frequency potion of the z¨1 signal is filtered and negated, analogous to the θ¨2 signal. For details, see [10].
5
Motion Scenarios for Mobile Robot
To test the effectiveness of the motion planning algorithm some sample input motion of the mobile
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Figure 7: Horizontal Scenario: Mobile Robot only moves in y1 -direction As illustrated in Figure 7, the robot moves horizontally with a maximum acceleration of 4 sm2 until it reaches a constant velocity. No changes in orientation occur, i.e. angle θ1 (t) = 0 . This scenario simulates the realistic case when the mobile robot accelerates from rest to a certain constant velocity. The deceleration scenario would have symmetric effects on the actuation of the parallel platform and is therefore not investigated explicitly.
5.2
Wave Scenario
The Wave Scenario provides a tool to test the response of the motion planning schemes for the parallel manipulator to a highly dynamic input. The upper graph in Figure 8 shows the motion of the mobile robot in the y1 − z1 plane: the mobile robot moves on a sinusoidal curve followed by a horizontal plateau. The remaining graphs in Figure 8 show the separate components, y1 , z1 , θ1 of the trajectory of the mobile robot over time. The mobile robot is simulated to move along this path with its orientation θ1 always tangential to it. The constant velocity v along the path, the frequency ω of the sinusoidal part, and the factor f controlling the amplitude can all be specified by the user.
6
Simulation Results
To assess the performance of different motion planning algorithms, the values of the residual forces (normal and friction) and reaction moments of the object on the PPM are compared to those without any actuation. The residual forces and moments are calculated from Equations (2), (3), (4) with the outputs of the motion planning algorithms. Dividing the specific normal
Plot of the Trajectory in YZ Plane
Horizontal Input Trajectory Results
0.03
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FMPA GMA OCA unactuated case
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scenario and the 3-DOF algorithms for Wave scenario are shown here. 0
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Figure 9: Comparison of 1-DOF Algorithms Results
−2 0
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Figure 8: Wave Scenario: Trajectory of Mobile Robot F force, N m , by the specific friction force, N , provides the required friction coefficient, µ, to keep the object from F > 0). This quantity characterizes sliding (assumed N the stability of the object from sliding. Therefore, the simulation results of µ, instead of F and N, are presented here. Due to space limitation, results for the reaction moment can be found in [10]. However, in general, it can be said that the trend for the reaction moment, Mreaction , follows that of µ. Therefore, by looking at µ, one can induce the trend in Mreaction . This is simply a rule of thumb induced from the data set. In addition to comparing the relative values of the residual forces and moments to assess performance of the different motion algorithms, these values can be compared with threshold values that would make the object on the PPM unstable. In practice, these threshold values depend on the dimensions and mass of the object, as well as the parameters of the surface on which it resides. Due to these uncertainties in object dimensions and surface parameters, only the stability for one set of fixed threshold values were explored. The results of these findings can be found in [10]. Simulation of the motion planning algorithms was done in the Matlab environment, with the Simulink Toolbox. Due to space limitations, only the results of the 1-DOF motion planning algorithms for Horizontal
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One-DOF Motion Planning Results
The results of the one-DOF motion planning algorithms are shown in Figure 9. These results are for the Horizontal scenario and only the angle of the PPM is actuated (no linear actuation). There are four sets of data in this figure. The dashed line indicates the required friction coefficient to keep the object stable on the PPM without any actuation (compensation turned off). The other lines show the performance of the FMPA, the GMA based on sine functions and the OCA algorithm, in reducing the required friction coefficient. In general, the FMPA online scheme has the lowest value of µ over the long periods of time. As is apparent from the figure, the motion planning scheme drastically improves the performance of the system, as measured by the level of stability of the object.
6.2
Three-DOF Motion Planning Results
A comparison of the required friction coefficient for the 3DOF-FMPA, washout filter and the unactuated case are compared in Figure 10. Notice the drastic improvement that the 3DOF-FMPA provides over the unactuated and washout filter schemes. The friction coefficient is close to zero until the mobile robot reaches the 10 second mark and has to transition from an uphill climb to a level terrain. This transition requires some level of static friction to help the object remain stable. This level is still significantly below the unactuated and washout filter case. For other scenarios the washout filter performed better than in the example shown in Figure 10. Furthermore, the performance of the washout filter depends heavily on the selection of the filter parameters and may leave room for improvement. Nevertheless, for the parameters chosen here, its results were always significantly worse than that of the 3DOF-FMPA.
Wave Input Trajectory Results
(Studienstiftung), both awarded to Michael Decker. In addition this research was supported by a doctoral fellowship of the Engineering Research Program of the Office of Basic Energy Sciences at the Department of Energy, awarded to Anh Dang.
0.25 W/O Actuation Washout Filter 3DOFFMPA
Required Friction Coeff
0.2
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References 0.1
[1] R. Graf and R. Dillmann, “Active acceleration compensation using a Stewart-platform on a mobile robot,” in Proceedings of the 2nd Euromicro Workshop on Advanced Mobile Robots, EUROBOT 97, (Brescia, Italy), pp. 59 – 64, 1997.
3DOFFMPA 0.05
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Figure 10: Comparison of 3-DOF Algorithm and Washout Filter Results
7
Conclusions and Future Work
Several strategies were proposed for active acceleration compensation to facilitate the transport of delicate objects. A performance measure for the system was developed on which all proposed motion planning algorithms were based. When considering the motion planning for the tilting of the parallel manipulator, both off-line (OCA & GMA) and online (FMPA) motion planning schemes were assessed. For the full three DOF actuating scheme, the 3DOF-FMPA was developed and compared with the motion planning algorithm called the washout filter. For the one-DOF Horizontal scenario case, simulation results showed that the actuation schemes of both the off-line and on-line motion planning algorithms improved the stability of an object on a PPM, and that the online motion planning scheme gave better results than the off-line schemes. Even better system performance was achieved by the full three-DOF actuation scheme, which out-performed the washout-filter motion planning scheme. Encouraged by the improvements that can be obtained even with a 1-DOF device, a prototype of a 1-DOF acceleration compensation device is currently being constructed at Georgia Tech. Future work includes the integration of sensors, experiments with the 1-DOF prototype and the construction of a 3-DOF prototype.
8
Acknowledgements
This work was supported by fellowships of the German Academic Exchange Service (DAAD) and the German National Merit Scholarship Foundation
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[2] R. Graf and R. Dillmann, “Aktive beschleunigungskompensation mittels einer Stewart-Plattform auf einem mobilen roboter (in German),” in Proceedings of Autonome Mobile Systeme - AMS 97, (Stuttgart, Germany), p. unknown, 1997. Available at http://wwwipr.ira.uka.de/publications/papers.html. [3] R. Graf and R. Dillmann, “Acceleration compensation using a Stewart-platform on a mobile robot,” in Proceedings of the 3rd Euromicro Workshop on Advanced Mobile Robots, EUROBOT 99, (Zuerich, Switzerland), pp. 17 – 24, Sept 6-8 1999. [4] R. Graf and R. Dillmann, “Die Stewart-Plattform als dynamisches lastaufnahmesystem eines mobilen roboters (in German),” in Proceedings of Autonome Mobile Systeme - AMS 99, (Munich, Germany), pp. 150 – 159, 1999. [5] R. Graf and R. Dillmann, “Ein flexibles lastaufnahmesystem - die Stewart-Plattform (in German),” in Robotik 2000, p. unknown, 2000. Available at http://wwwipr.ira.uka.de/publications/papers.html. [6] M. Nahon and L. Reid, “Simulator motion-drive algorithms. a designer’s perspective,” Journal of Guidance, Control, and Dynamics, vol. 13, pp. 356– 362, March–April 1990. [7] L. Reid and M. Nahon, “Flight simulation motionbase drive algorithms. part 1: Developing and testing the equations.,” Tech. Rep. UTIAS 296PT1, Toronto University, Inst. for Aerospace Studies, December 1985. [8] L. Reid and M. Nahon, “Flight simulation motion-base drive algorithms. part 2: Selecting the system parameters.,” Tech. Rep. UTIAS 307, Toronto University, May 1986. [9] M. Baarspul, “Lecture notes on flight simulation techniques,” Tech. Rep. LR-596, Technische Hogeschool Delft (Netherlands). Faculty of Aerospace Engineering, 1989. [10] M. Decker, “Active acceleration compensation for transport of delicate objects,” Master’s thesis, Georgia Institute of Technology, Atlanta, Georgia, August 2000.
