Hybrid System Behavior Speci cation for Multiple Robotic Mechanisms
This paper is published in the Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems IROS, pp. 140-147, Osaka, Japan, 1996.
Hybrid System Behavior Speci cation for Multiple Robotic Mechanisms Martin Buss
Gunther Schmidt
Department of Automatic Control Engineering Technical University of Munich D-80290 Munchen, Germany E-mail: martin/[email protected]
In this paper we propose a novel approach to reference behavior speci cation for multiple robotic mechanisms using hybrid system models. Hybrid automata can specify discrete states (operational modes) and continuous variable reference values in one uni ed framework. An example mechanism with 3 degrees-offreedom and a 6-legged walking machine with a combination of several hybrid automata for reference generation and synchronization are discussed. Simulation results show that a hybrid system model is an e ective method for robotic behavior speci cation. Using a model veri cation tool we show that behavior correctness veri cation and parametric analysis are possible.
1 Introduction
Multiple robotic mechanisms have been investigated to realize more exible and complex behavior otherwise not achievable with single robots. Active research areas are cooperating robot manipulators which may carry heavy loads [1], dextrous multi ngered hands which are used for complex manipulation tasks [2{6] and walking machines which may traverse rougher terrain than conventional wheel-based mobile robots [7,8]. Higher degrees of exibility are achieved by redundancy in the system and intelligent control strategies are being proposed. Speci c control strategies include e.g. stable, rolling or sliding contact states of dextrous hands [9{15], heuristic biologically motivated walking machine control [8], planning and learning schemes for motion trajectories [16{18], hybrid position/force or sti ness control schemes [19{22]. However, it remains dicult to model the entire behavior of multiple mechanisms in a more global and integrated way, when the operation state changes among isolated control strategies.
In this paper we propose a hybrid system approach as a step toward more exible and global behavior speci cation models for multiple mechanisms. The novel idea is to model the reference behavior to be realized by appropriate controllers. A related idea taken as the basis for this approach is to model complex manipulation tasks in motion skill models [16], not considering discrete states. Figure 1 illustrates the idea of simultaneous reference speci cation for the two parts of a hybrid process, the discrete event dynamical system|DEDS and continuous value dynamical system|CVDS parts.
reference behavior DEDS CVDS
hybrid automaton
Abstract
events Hybrid Control
DEDS
CVDS Control
CVDS
hybrid process
continuous state values Figure 1: Hybrid system reference generation and control. A hybrid system model for DEDS/CVDS reference generation is shown in an example of a simple 3 degrees-of-freedom (DOF) robotic mechanism. Speci-
fying discrete state (operation/control mode) and continuous reference values simultaneously enables improved control behavior during operation mode traversal. This is demonstrated by simulation results where the control mode switches between trajectory following and force control. The new approach is compared to a conventional method based on kinematic conditions for mode switching. Next, the hybrid system approach is applied to a more complicated example of a walking machine. Two leg groups and a synchronizing supervisor hybrid automaton are given. Compared to other approaches, e.g. a temporal logic based controller synthesis [23], we believe that the combination of several hybrid automata provides a better model of walking, especially with respect to the CVDS part. The behavior of the combined automaton can be analyzed using HyTech developed by Henzinger et al. [24{28]. In our example we use HyTech to analyze the combined hybrid automaton with respect to controller correctness and also perform some parameter analysis. The rest of the paper is structured as follows: Section 2 discusses the example of the 3-DOF mechanism and the trajectory/operation mode generating hybrid system model. The hybrid system approach applied to a walking machine with a synchronizing supervisory controller is presented in Section 3 and Section 4 discusses some simulation and parametric analysis results.
2 Hybrid System Model
For multiple robotic mechanisms with several kinematic chains hybrid system models are developed. Hybrid systems comprise a discrete event model most often a state automaton and a model of the continuous behavior. Contact state of a particular chain with the xed environment forms the DEDS part of the model and continuous reference values the CVDS part. Combined speci cation of the behavior for the mechanism is possible using a hybrid automaton. Hybrid automata specify in each discrete state the time-derivatives of the corresponding continuous state variables. Restricting the continuous state timederivatives to constants yields the class of linear hybrid automata with several advantages to automatic veri cation and parametric analysis [24{28].
2.1 Example: 3-DOF mechanism
Figure 2 shows as an example a mechanism with 3 DOF. Several of these can be combined to a multiple mechanism similar to dextrous hands or walking
machines. Further, let us consider as an example the desired path for the tip shown in Figure 3. The path is a rectangle in the y=z ?plane at a certain distance x0 in the workspace of the mechanism. Considering a rectangular path is for simplicity of the discussion but without loss of generality. More complex paths can be approximated by polygons within the same class of linear hybrid automata, however, the number of discrete states will increase significantly. z
y x
1
2
3
Figure 2: 3 DOF kinematic chain.
SWING zl zc
z PUT x y fz SLIDE yl
LIFT
yr
Figure 3: Example rectangular path for the tip. The considered rectangular reference path has 4 discrete states: LIFT{SWING{PUT{SLIDE. During the SLIDE-state the controller switches operation modes to force control in z ?direction, while all other directions are position controlled. Including time a reference trajectory can be obtained by a hybrid automaton, see Figure 4. In each discrete state the behavior for the continuous state variables x; y; z (tip position in cartesian space, x = x0 at all times) is speci ed together with an invariant condition. When the invariant condition no longer is satis ed, i.e. an event occurs, a transition to the next state takes place.
SWING y yl y_ = vswing z_ = 0
x = x0 y = yr z=0 z
y
= zl
= yl
= zc
z
LIFT z zl y_ = 0 z_ = vlift
y
PUT z zc y_ = 0 z_ = vput
= yr
SLIDE y yr y_ = vslide z_ = 0
Figure 4: Hybrid system reference trajectory generator. The parameters for this hybrid automaton are the swing forward, approach, slide and lift velocities of the tip, v < 0, v < 0, v > 0, v > 0, respectively; z , z , y , y describe the path geometry as depicted in Figure 3. Remark 1 The invariants z z and z z can be modi ed to include e.g. contact sensor information z z ^ s with s a boolean contact sensor signal. Remark 2 A distinct advantage of this hybrid system model is that the continuous control strategy for the mechanism can be switched easily to force control during the discrete state SLIDE. Corresponding simulation results are discussed in Section 4.1. swing
l
put
c
l
slide
r
c
c
contact
lif t
state STABLE, its contact point location is given by the continuous state x. The continuous variable w weighs the contribution of this nger to the overall grasp: w = 1 if the nger is of equal importance as the other N ? 1 ngers; w = 0 when the nger contact force approaches zero. When the regrasping sequence is commanded by a signal rs the automaton changes state into REDUCE, where w is gradually taken to zero with the rate > 0 (see Section 3 for an explanation of synchronizing signals like rs). Following is the LIFT/MOVE state which takes the nger to the new contact location. The state INCREASE gradually increases w and when w = 1 a stable grasp is achieved again. Remark 3 The actual implementation of the hand controller is straightforward from this hybrid automaton. We use the grasping force optimization algorithm proposed earlier [29,30] with an additional linear constraint (depending on w) on the normal force of the lift-o nger. We have promising results when combining this approach with a sti ness control scheme [31]. However, due to space constraints these results will be presented elsewhere.
l
contact
2.2 Example: dextrous hand regrasping
This second example of dextrous hand regrasping uses another continuous variable in the hybrid automaton to reduce/increase the contact force contribution of one particular nger. Consider a multi- ngered hand with N 3 ngers grasping an object stably. The task goal of regrasping is to lift one nger o the object, move it to another contact point and remake contact. We assume that if the nger breaks contact with the object the resulting grasp with N ? 1 ngers is still stable. To realize this task of regrasping, the contact force of the nger should be gradually reduced until lift-o . We can use the hybrid automaton shown in Figure 5 to specify this regrasping task. The gure shows only the part for the lift-o nger of the complete automaton. Assuming a stable grasp with the nger in the
STABLE
x_ = 0
REDUCE w0
rs
x_ = 0
w_ = ?
w_ = 0 w=1 INCREASE w1
x_ = 0
w_ =
w w=1
=0 LIFT/MOVE scontact x_ = f (x; t) w_ = 0
Figure 5: Hybrid automaton for regrasping showing only 1 of N ngers.
2.3 Variable Structure CVDS Control
Depending on the discrete state of the hybrid automaton a di erent control law may be required. For the 3-DOF mechanism in Section 2.1 during the state SLIDE a force control law is needed and a
signal1
y_ = 0 z_ = 0 y
LIFT z zl y_ = 0 z_ = vlift
z
= yr
SLIDE y yr y_ = vslide z_ = 0
signal2
SWING y yl y_ = vswing z_ = 0
= zl
y SLB y_ = 0 z_ = 0
z
= yl PUT z zc y_ = 0 z_ = vput
= zc
Figure 6: Hybrid system reference trajectory generator for leg groups of a 6-legged walking machine. position controller otherwise. Dextrous regrasping (see Section 2.2) requires a force controller in state STABLE, a force planning algorithm in states REDUCE and INCREASE and a position control law in state LIFT/MOVE. The control law selection strategy therewith depends on the discrete state as well as the selection of the continuous reference variable subset of the hybrid automaton. A systematic way to design such hybrid control systems is an important research goal and topic of our current work. Some results how to generate supervisory controllers are presented in the next Section 3. However, currently we are lacking a desirable formal design procedure for supervisors and the controller switching strategy.
3 Supervisory Controller The hybrid automaton shown above has no possibility of synchronization or control. Inserting two \waiting" states and synchonization events signal1, signal2 yields the extended hybrid automaton shown in Figure 6 with the new states SLE (end of slide) and SLB (begin of slide), where all time-derivatives of the state variables are zero and transition takes place through the synchronized events. Now, consider a 6-legged walking machine with two groups 1 and 2 of legs, see Figure 8. For each group a reference generating hybrid system as shown in Fig-
2 1 2
G1
t_ = 1 signal1
SLE
signal2
x = x0 y = yr z = zc
G2 t_ = 1
Figure 7: Walking machine supervisor.
body
1
y
x
2 1
Figure 8: Six-legged walking machine with two leg groups. ure 6 is used, where the automaton for group 2 is di erent only in that the beginning state is SLB and the synchronization events are to be exchanged. Goals for a supervisory controller are the correct synchronization of the two groups of legs such that at least one group remains in contact with the environment at all times. Further by adjusting the parameters in the invariants (trajectory geometry) a larger/smaller step, or higher/lower lift can be achieved. Of particular interest may be also the forward or turning speed. In this example the supervision task is simple and the problem of formal supervisor synthesis is not addressed here. Formal supervisor synthesis for more complex settings is an issue of our current research.
The supervisor hybrid automaton is shown in Figure 7, containing only two states G1 and G2 . The timederivative of the time variable t is always 1. It is clear that this supervisor results in correct behavior: while leg group 1 swings left to right, leg group 2 drives the walking machine forward. Synchronization takes place in states SLB, SLE and then the leg groups change role. State G of the supervisor means that leg group i is driving (sliding). This simple model with leg groups can be extended to a more complicated and more exible one when each leg is modelled by its own hybrid automaton and the supervisor synchronizes all legs according to the behavior goal of e.g. straight or curve walking. For such a combined automaton the supervisor synthesis is more complex and practically impossible without a formal design procedure. Further, it turned out that HyTech could not build the resulting product automaton due to resource constraints. For the simple case of two leg groups we can automatically verify system correctness and some parameter analysis is possible using HyTech, discussed in the following Section 4.2. i
Remark 4 To clarify the general hybrid system and
control structure of Figure 1 to this special example: the hybrid automata (one for each leg group) are represented by the DEDS/CVDS reference behavior speci cation block and the synchronizing supervisor together with the control law switching strategy form the hybrid control block.
4 Simulations
4.1 Hybrid Model Based Contact Behavior
A dynamic model of the 3-DOF mechanism shown in Figure 2 together with the reference generating hybrid automaton of Figure 4 was simulated. The cartesian control strategy is a hybrid position/force control law as
= J (K T
f
ef
+K
p
ep + K v e_ p )
(1)
where e = f ? f denotes the force error with the desired force f , e = x ? x, e_ = e the position and velocity errors, x = [ x y z ] the desired position speci ed by the hybrid automaton and x = [ x y z ] the actual tip position, both in cartesian space. J denotes the transpose of the Jacobian and the joint torques. The force, position and velocity gain matrices K , K , K are switched according to the discrete state of the hybrid automaton. During the SLIDE state force control in z ?direction is used with d
f
d
d
p
d
T
T
f
p
d
p
d
v
d
dt d
p T
a nominal force of f = ?5 N and position control for all other operation states. The mechanism joint length is l1 = 0:055m, l2 = 0:25m, l3 = 0:325m, the trajectory geometry parameters are z = 0:05m, z = ?0:05m, y = ?0:05m, y = 0:05m and the velocities are v = v = ?0:3m=s, v = v = 0:3m=s. Figure 9 shows the discrete state of the hybrid automaton coded as SWING=1, PUT=3, SLIDE=4, LIFT=5. The reference and simulated path of the tip are shown in Figure 10. Using the hybrid automaton state to switch between force control during SLIDE and position control in all other states yields a smooth contact force behavior, see Figure 11. Control modes can alternatively be switched based on the evaluation of a kinematic condition on the z ?position error e = z ? z where z is the desired value from the hybrid automaton reference value, such that d z
l
c
l
swing
slide
r
put
lif t
d
z
d
0:0007 e 0:002 ! force control otherwise ! position control z
where during position control the gain matrices are K = diag(0; 0; 1), K = k diag(1; 1; 0), and during force control K = 0, K = k I 3 ; diag() denotes the diagonal matrix with the elements in the diagonal, I the n n unity matrix and k > 0, k > 0 scalar position and velocity gains. This yields approximately the same trajectory behavior, however, the force behavior during contact with the environment is non-smooth and quite undesirable, see Figure 12. This is caused by rapid controller switching, i.e. the kinematic constraint is triggered rapidly by dynamic e ects. f
p;v
f
p;v
p;v
p;v
n
p
v
Discrete state 6 LIFT
4 2 0 0
SLIDE PUT SWING
SWING 0.5
time [s]
1
1.5
Figure 9: Discrete state.
4.2 Model Veri cation and Parameter Analysis
Controller correctness in the sense that at least one leg group of the 6-legged walking machine (see Section 3) remains in contact with the environment could
0.1
10
z [m]
0.05
t=0 reference path simulated path
0
−0.05
−fz [N]
t=1.5
t=1.0
−fzd
5
t=0.5
−0.1 −0.06
−0.04
−0.02
0 y [m]
0.02
0.04
0.06
Figure 10: Path of the tip.
0 0
0.5
1
time [s]
1.5
Figure 12: Normal force f using the conventional switching scheme based on a kinematic condition. z
10
be veri ed using HyTech [27]. We considered the three hybrid automata leg1 , leg2 together with the synchronizing supervisor of Figure 7 and the safety requirement that the two leg groups should not be in the SWING, PUT or LIFT state simultaneously, i.e. the intersection of the safety region (leg1 = LIFT _ leg1 = SWING _ leg1 = PUT) ^ (leg2 = LIFT _ leg2 = SWING _ leg2 = PUT) with the reachable region of the hybrid automaton should be empty. The result obtained by HyTech is that the region speci ed by the safety requirement cannot be reached from the initial conditions. The conclusion then is that the supervisor is correct in that it guarantees the correct synchronization of the two leg groups. Parametric analysis could also be achieved using HyTech. We simply assume that the walking machine travels the distance covered by a leg group during the SLIDE state. Further, we specify that the parameters y , y for the leg groups should be equal for both, but be free parameters. Accumulating the distance covered by the walking machine in an additional continuous variable p, the safety requirement y1 ?0:05 _ y1 0:05 _ y2 ?0:05 _ y2 0:05 _ (p 0:5 ^ t = 10) was veri ed. This meaning that the walking machine should move forward more than 0:5m in 10s. The
A novel approach using hybrid automata for simultaneous discrete state (control mode) and trajectory reference generation for multiple robotic mechanisms has been presented. An example of a 3-DOF robotic mechanism has illustrated that the approach is effective. Simulations show that control switching is successful and that force control during environment contact is smooth. The behavior was compared to a conventional control switching strategy based on kinematic conditions, which yields undesirable force control characteristics. Application to a 6-legged walking mechanism with two groups of three legs has shown that hybrid automata can give more insight to the overall behavior of the system. Advantages are mainly through increased accuracy of the continuous part of the model. A hybrid supervisor combined with two hybrid automata for the two leg groups was veri ed using the tool HyTech in that correct synchronization behavior of the leg groups was proofed. Parametric analysis of the combined hybrid model automatically yields valid parameter domains. Another approach for supervisory controller synthesis of walking machines using temporal logic [23] does not provide these possibilities of veri cation and analysis of the continuous behavior. Our conclusions are that hybrid automata are a viable approach to formalized speci cation of more exible and global behavior models of multiple robotic mechanisms. Further, automatic veri cation and
−fz [N]
Figure 11: Normal force f using the hybrid automaton.
result of the parameter range for violation of the safety requirement is y ?0:05 _ y 0:05 _ y ?0:05 _ y 0:05 _ y y + 0:05 which means that the step length should be at least 0.05m.
−fzd
5
0 0
0.5
time [s]
1
1.5
l
l
r
r
r
z
l
r
5 Conclusion
l
parametric analysis are possible to some extend. Shortcomings of the proposed approach are that the complexity of veri cation and analysis is not polynomial. Convergence to the reachable regions in the state space of hybrid automata is often dicult to achieve [27, 28]. This seems especially to be the case for parametric analysis with several free tunable parameters in the hybrid system model. Future research has to concentrate on formal supervisor synthesis within the framework of hybrid automata. Inclusion of ideas from Ramadge-Wonham supervisory control theory for DEDS [32] modi ed to the domain of hybrid automata may give important insights to formal synthesis. Our current research includes this aspect and the application of the presented approach to dextrous multi- ngered robotic hands.
[8]
[9]
[10]
Acknowledgments
This work was partly supported by the German National Science Foundation (DFG) in the frame work of the KONDISK project.
[11]
References
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Hybrid System Behavior Speci cation for Multiple Robotic Mechanisms Martin Buss
Gunther Schmidt
Department of Automatic Control Engineering Technical University of Munich D-80290 Munchen, Germany E-mail: martin/[email protected]
In this paper we propose a novel approach to reference behavior speci cation for multiple robotic mechanisms using hybrid system models. Hybrid automata can specify discrete states (operational modes) and continuous variable reference values in one uni ed framework. An example mechanism with 3 degrees-offreedom and a 6-legged walking machine with a combination of several hybrid automata for reference generation and synchronization are discussed. Simulation results show that a hybrid system model is an e ective method for robotic behavior speci cation. Using a model veri cation tool we show that behavior correctness veri cation and parametric analysis are possible.
1 Introduction
Multiple robotic mechanisms have been investigated to realize more exible and complex behavior otherwise not achievable with single robots. Active research areas are cooperating robot manipulators which may carry heavy loads [1], dextrous multi ngered hands which are used for complex manipulation tasks [2{6] and walking machines which may traverse rougher terrain than conventional wheel-based mobile robots [7,8]. Higher degrees of exibility are achieved by redundancy in the system and intelligent control strategies are being proposed. Speci c control strategies include e.g. stable, rolling or sliding contact states of dextrous hands [9{15], heuristic biologically motivated walking machine control [8], planning and learning schemes for motion trajectories [16{18], hybrid position/force or sti ness control schemes [19{22]. However, it remains dicult to model the entire behavior of multiple mechanisms in a more global and integrated way, when the operation state changes among isolated control strategies.
In this paper we propose a hybrid system approach as a step toward more exible and global behavior speci cation models for multiple mechanisms. The novel idea is to model the reference behavior to be realized by appropriate controllers. A related idea taken as the basis for this approach is to model complex manipulation tasks in motion skill models [16], not considering discrete states. Figure 1 illustrates the idea of simultaneous reference speci cation for the two parts of a hybrid process, the discrete event dynamical system|DEDS and continuous value dynamical system|CVDS parts.
reference behavior DEDS CVDS
hybrid automaton
Abstract
events Hybrid Control
DEDS
CVDS Control
CVDS
hybrid process
continuous state values Figure 1: Hybrid system reference generation and control. A hybrid system model for DEDS/CVDS reference generation is shown in an example of a simple 3 degrees-of-freedom (DOF) robotic mechanism. Speci-
fying discrete state (operation/control mode) and continuous reference values simultaneously enables improved control behavior during operation mode traversal. This is demonstrated by simulation results where the control mode switches between trajectory following and force control. The new approach is compared to a conventional method based on kinematic conditions for mode switching. Next, the hybrid system approach is applied to a more complicated example of a walking machine. Two leg groups and a synchronizing supervisor hybrid automaton are given. Compared to other approaches, e.g. a temporal logic based controller synthesis [23], we believe that the combination of several hybrid automata provides a better model of walking, especially with respect to the CVDS part. The behavior of the combined automaton can be analyzed using HyTech developed by Henzinger et al. [24{28]. In our example we use HyTech to analyze the combined hybrid automaton with respect to controller correctness and also perform some parameter analysis. The rest of the paper is structured as follows: Section 2 discusses the example of the 3-DOF mechanism and the trajectory/operation mode generating hybrid system model. The hybrid system approach applied to a walking machine with a synchronizing supervisory controller is presented in Section 3 and Section 4 discusses some simulation and parametric analysis results.
2 Hybrid System Model
For multiple robotic mechanisms with several kinematic chains hybrid system models are developed. Hybrid systems comprise a discrete event model most often a state automaton and a model of the continuous behavior. Contact state of a particular chain with the xed environment forms the DEDS part of the model and continuous reference values the CVDS part. Combined speci cation of the behavior for the mechanism is possible using a hybrid automaton. Hybrid automata specify in each discrete state the time-derivatives of the corresponding continuous state variables. Restricting the continuous state timederivatives to constants yields the class of linear hybrid automata with several advantages to automatic veri cation and parametric analysis [24{28].
2.1 Example: 3-DOF mechanism
Figure 2 shows as an example a mechanism with 3 DOF. Several of these can be combined to a multiple mechanism similar to dextrous hands or walking
machines. Further, let us consider as an example the desired path for the tip shown in Figure 3. The path is a rectangle in the y=z ?plane at a certain distance x0 in the workspace of the mechanism. Considering a rectangular path is for simplicity of the discussion but without loss of generality. More complex paths can be approximated by polygons within the same class of linear hybrid automata, however, the number of discrete states will increase significantly. z
y x
1
2
3
Figure 2: 3 DOF kinematic chain.
SWING zl zc
z PUT x y fz SLIDE yl
LIFT
yr
Figure 3: Example rectangular path for the tip. The considered rectangular reference path has 4 discrete states: LIFT{SWING{PUT{SLIDE. During the SLIDE-state the controller switches operation modes to force control in z ?direction, while all other directions are position controlled. Including time a reference trajectory can be obtained by a hybrid automaton, see Figure 4. In each discrete state the behavior for the continuous state variables x; y; z (tip position in cartesian space, x = x0 at all times) is speci ed together with an invariant condition. When the invariant condition no longer is satis ed, i.e. an event occurs, a transition to the next state takes place.
SWING y yl y_ = vswing z_ = 0
x = x0 y = yr z=0 z
y
= zl
= yl
= zc
z
LIFT z zl y_ = 0 z_ = vlift
y
PUT z zc y_ = 0 z_ = vput
= yr
SLIDE y yr y_ = vslide z_ = 0
Figure 4: Hybrid system reference trajectory generator. The parameters for this hybrid automaton are the swing forward, approach, slide and lift velocities of the tip, v < 0, v < 0, v > 0, v > 0, respectively; z , z , y , y describe the path geometry as depicted in Figure 3. Remark 1 The invariants z z and z z can be modi ed to include e.g. contact sensor information z z ^ s with s a boolean contact sensor signal. Remark 2 A distinct advantage of this hybrid system model is that the continuous control strategy for the mechanism can be switched easily to force control during the discrete state SLIDE. Corresponding simulation results are discussed in Section 4.1. swing
l
put
c
l
slide
r
c
c
contact
lif t
state STABLE, its contact point location is given by the continuous state x. The continuous variable w weighs the contribution of this nger to the overall grasp: w = 1 if the nger is of equal importance as the other N ? 1 ngers; w = 0 when the nger contact force approaches zero. When the regrasping sequence is commanded by a signal rs the automaton changes state into REDUCE, where w is gradually taken to zero with the rate > 0 (see Section 3 for an explanation of synchronizing signals like rs). Following is the LIFT/MOVE state which takes the nger to the new contact location. The state INCREASE gradually increases w and when w = 1 a stable grasp is achieved again. Remark 3 The actual implementation of the hand controller is straightforward from this hybrid automaton. We use the grasping force optimization algorithm proposed earlier [29,30] with an additional linear constraint (depending on w) on the normal force of the lift-o nger. We have promising results when combining this approach with a sti ness control scheme [31]. However, due to space constraints these results will be presented elsewhere.
l
contact
2.2 Example: dextrous hand regrasping
This second example of dextrous hand regrasping uses another continuous variable in the hybrid automaton to reduce/increase the contact force contribution of one particular nger. Consider a multi- ngered hand with N 3 ngers grasping an object stably. The task goal of regrasping is to lift one nger o the object, move it to another contact point and remake contact. We assume that if the nger breaks contact with the object the resulting grasp with N ? 1 ngers is still stable. To realize this task of regrasping, the contact force of the nger should be gradually reduced until lift-o . We can use the hybrid automaton shown in Figure 5 to specify this regrasping task. The gure shows only the part for the lift-o nger of the complete automaton. Assuming a stable grasp with the nger in the
STABLE
x_ = 0
REDUCE w0
rs
x_ = 0
w_ = ?
w_ = 0 w=1 INCREASE w1
x_ = 0
w_ =
w w=1
=0 LIFT/MOVE scontact x_ = f (x; t) w_ = 0
Figure 5: Hybrid automaton for regrasping showing only 1 of N ngers.
2.3 Variable Structure CVDS Control
Depending on the discrete state of the hybrid automaton a di erent control law may be required. For the 3-DOF mechanism in Section 2.1 during the state SLIDE a force control law is needed and a
signal1
y_ = 0 z_ = 0 y
LIFT z zl y_ = 0 z_ = vlift
z
= yr
SLIDE y yr y_ = vslide z_ = 0
signal2
SWING y yl y_ = vswing z_ = 0
= zl
y SLB y_ = 0 z_ = 0
z
= yl PUT z zc y_ = 0 z_ = vput
= zc
Figure 6: Hybrid system reference trajectory generator for leg groups of a 6-legged walking machine. position controller otherwise. Dextrous regrasping (see Section 2.2) requires a force controller in state STABLE, a force planning algorithm in states REDUCE and INCREASE and a position control law in state LIFT/MOVE. The control law selection strategy therewith depends on the discrete state as well as the selection of the continuous reference variable subset of the hybrid automaton. A systematic way to design such hybrid control systems is an important research goal and topic of our current work. Some results how to generate supervisory controllers are presented in the next Section 3. However, currently we are lacking a desirable formal design procedure for supervisors and the controller switching strategy.
3 Supervisory Controller The hybrid automaton shown above has no possibility of synchronization or control. Inserting two \waiting" states and synchonization events signal1, signal2 yields the extended hybrid automaton shown in Figure 6 with the new states SLE (end of slide) and SLB (begin of slide), where all time-derivatives of the state variables are zero and transition takes place through the synchronized events. Now, consider a 6-legged walking machine with two groups 1 and 2 of legs, see Figure 8. For each group a reference generating hybrid system as shown in Fig-
2 1 2
G1
t_ = 1 signal1
SLE
signal2
x = x0 y = yr z = zc
G2 t_ = 1
Figure 7: Walking machine supervisor.
body
1
y
x
2 1
Figure 8: Six-legged walking machine with two leg groups. ure 6 is used, where the automaton for group 2 is di erent only in that the beginning state is SLB and the synchronization events are to be exchanged. Goals for a supervisory controller are the correct synchronization of the two groups of legs such that at least one group remains in contact with the environment at all times. Further by adjusting the parameters in the invariants (trajectory geometry) a larger/smaller step, or higher/lower lift can be achieved. Of particular interest may be also the forward or turning speed. In this example the supervision task is simple and the problem of formal supervisor synthesis is not addressed here. Formal supervisor synthesis for more complex settings is an issue of our current research.
The supervisor hybrid automaton is shown in Figure 7, containing only two states G1 and G2 . The timederivative of the time variable t is always 1. It is clear that this supervisor results in correct behavior: while leg group 1 swings left to right, leg group 2 drives the walking machine forward. Synchronization takes place in states SLB, SLE and then the leg groups change role. State G of the supervisor means that leg group i is driving (sliding). This simple model with leg groups can be extended to a more complicated and more exible one when each leg is modelled by its own hybrid automaton and the supervisor synchronizes all legs according to the behavior goal of e.g. straight or curve walking. For such a combined automaton the supervisor synthesis is more complex and practically impossible without a formal design procedure. Further, it turned out that HyTech could not build the resulting product automaton due to resource constraints. For the simple case of two leg groups we can automatically verify system correctness and some parameter analysis is possible using HyTech, discussed in the following Section 4.2. i
Remark 4 To clarify the general hybrid system and
control structure of Figure 1 to this special example: the hybrid automata (one for each leg group) are represented by the DEDS/CVDS reference behavior speci cation block and the synchronizing supervisor together with the control law switching strategy form the hybrid control block.
4 Simulations
4.1 Hybrid Model Based Contact Behavior
A dynamic model of the 3-DOF mechanism shown in Figure 2 together with the reference generating hybrid automaton of Figure 4 was simulated. The cartesian control strategy is a hybrid position/force control law as
= J (K T
f
ef
+K
p
ep + K v e_ p )
(1)
where e = f ? f denotes the force error with the desired force f , e = x ? x, e_ = e the position and velocity errors, x = [ x y z ] the desired position speci ed by the hybrid automaton and x = [ x y z ] the actual tip position, both in cartesian space. J denotes the transpose of the Jacobian and the joint torques. The force, position and velocity gain matrices K , K , K are switched according to the discrete state of the hybrid automaton. During the SLIDE state force control in z ?direction is used with d
f
d
d
p
d
T
T
f
p
d
p
d
v
d
dt d
p T
a nominal force of f = ?5 N and position control for all other operation states. The mechanism joint length is l1 = 0:055m, l2 = 0:25m, l3 = 0:325m, the trajectory geometry parameters are z = 0:05m, z = ?0:05m, y = ?0:05m, y = 0:05m and the velocities are v = v = ?0:3m=s, v = v = 0:3m=s. Figure 9 shows the discrete state of the hybrid automaton coded as SWING=1, PUT=3, SLIDE=4, LIFT=5. The reference and simulated path of the tip are shown in Figure 10. Using the hybrid automaton state to switch between force control during SLIDE and position control in all other states yields a smooth contact force behavior, see Figure 11. Control modes can alternatively be switched based on the evaluation of a kinematic condition on the z ?position error e = z ? z where z is the desired value from the hybrid automaton reference value, such that d z
l
c
l
swing
slide
r
put
lif t
d
z
d
0:0007 e 0:002 ! force control otherwise ! position control z
where during position control the gain matrices are K = diag(0; 0; 1), K = k diag(1; 1; 0), and during force control K = 0, K = k I 3 ; diag() denotes the diagonal matrix with the elements in the diagonal, I the n n unity matrix and k > 0, k > 0 scalar position and velocity gains. This yields approximately the same trajectory behavior, however, the force behavior during contact with the environment is non-smooth and quite undesirable, see Figure 12. This is caused by rapid controller switching, i.e. the kinematic constraint is triggered rapidly by dynamic e ects. f
p;v
f
p;v
p;v
p;v
n
p
v
Discrete state 6 LIFT
4 2 0 0
SLIDE PUT SWING
SWING 0.5
time [s]
1
1.5
Figure 9: Discrete state.
4.2 Model Veri cation and Parameter Analysis
Controller correctness in the sense that at least one leg group of the 6-legged walking machine (see Section 3) remains in contact with the environment could
0.1
10
z [m]
0.05
t=0 reference path simulated path
0
−0.05
−fz [N]
t=1.5
t=1.0
−fzd
5
t=0.5
−0.1 −0.06
−0.04
−0.02
0 y [m]
0.02
0.04
0.06
Figure 10: Path of the tip.
0 0
0.5
1
time [s]
1.5
Figure 12: Normal force f using the conventional switching scheme based on a kinematic condition. z
10
be veri ed using HyTech [27]. We considered the three hybrid automata leg1 , leg2 together with the synchronizing supervisor of Figure 7 and the safety requirement that the two leg groups should not be in the SWING, PUT or LIFT state simultaneously, i.e. the intersection of the safety region (leg1 = LIFT _ leg1 = SWING _ leg1 = PUT) ^ (leg2 = LIFT _ leg2 = SWING _ leg2 = PUT) with the reachable region of the hybrid automaton should be empty. The result obtained by HyTech is that the region speci ed by the safety requirement cannot be reached from the initial conditions. The conclusion then is that the supervisor is correct in that it guarantees the correct synchronization of the two leg groups. Parametric analysis could also be achieved using HyTech. We simply assume that the walking machine travels the distance covered by a leg group during the SLIDE state. Further, we specify that the parameters y , y for the leg groups should be equal for both, but be free parameters. Accumulating the distance covered by the walking machine in an additional continuous variable p, the safety requirement y1 ?0:05 _ y1 0:05 _ y2 ?0:05 _ y2 0:05 _ (p 0:5 ^ t = 10) was veri ed. This meaning that the walking machine should move forward more than 0:5m in 10s. The
A novel approach using hybrid automata for simultaneous discrete state (control mode) and trajectory reference generation for multiple robotic mechanisms has been presented. An example of a 3-DOF robotic mechanism has illustrated that the approach is effective. Simulations show that control switching is successful and that force control during environment contact is smooth. The behavior was compared to a conventional control switching strategy based on kinematic conditions, which yields undesirable force control characteristics. Application to a 6-legged walking mechanism with two groups of three legs has shown that hybrid automata can give more insight to the overall behavior of the system. Advantages are mainly through increased accuracy of the continuous part of the model. A hybrid supervisor combined with two hybrid automata for the two leg groups was veri ed using the tool HyTech in that correct synchronization behavior of the leg groups was proofed. Parametric analysis of the combined hybrid model automatically yields valid parameter domains. Another approach for supervisory controller synthesis of walking machines using temporal logic [23] does not provide these possibilities of veri cation and analysis of the continuous behavior. Our conclusions are that hybrid automata are a viable approach to formalized speci cation of more exible and global behavior models of multiple robotic mechanisms. Further, automatic veri cation and
−fz [N]
Figure 11: Normal force f using the hybrid automaton.
result of the parameter range for violation of the safety requirement is y ?0:05 _ y 0:05 _ y ?0:05 _ y 0:05 _ y y + 0:05 which means that the step length should be at least 0.05m.
−fzd
5
0 0
0.5
time [s]
1
1.5
l
l
r
r
r
z
l
r
5 Conclusion
l
parametric analysis are possible to some extend. Shortcomings of the proposed approach are that the complexity of veri cation and analysis is not polynomial. Convergence to the reachable regions in the state space of hybrid automata is often dicult to achieve [27, 28]. This seems especially to be the case for parametric analysis with several free tunable parameters in the hybrid system model. Future research has to concentrate on formal supervisor synthesis within the framework of hybrid automata. Inclusion of ideas from Ramadge-Wonham supervisory control theory for DEDS [32] modi ed to the domain of hybrid automata may give important insights to formal synthesis. Our current research includes this aspect and the application of the presented approach to dextrous multi- ngered robotic hands.
[8]
[9]
[10]
Acknowledgments
This work was partly supported by the German National Science Foundation (DFG) in the frame work of the KONDISK project.
[11]
References
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