Rolling Manipulation with a Single Control - Semantic Scholar
Rolling Manipulation with a Single Control Prasun Choudhury and Kevin Lynch Laboratory for Intelligent Mechanical Systems Department of Mechanical Engineering Northwestern University Evanston, IL 60208
Abstract | Underactuated manipulation is the process of controlling several object degrees-of-freedom with fewer robot actuators. Underactuated manipulation allows us to build dextrous robots with only a few actuators. In this paper we explore the possibility of useful dynamic manipulation with only a single actuator. Our case study is a ball rolling in an asymmetrical bowl which can be accelerated along one linear degree-of-freedom. For a real physical system we show that it is possible to control the equilibrium orientation of the ball on the three-dimensional manifold SO(3) using the single input. We have built an experimenFig. 1. The rolling system tal demonstration of this system and we have constructed a motion planner to nd a sequence of motions of the bowl to accomplish a desired reorientation. to an integral curve of the control vector eld and would be of little interest. Because of the existence of a drift vector I. Introduction eld, however, it is possible to take Lie brackets between f Underactuated robotic manipulation is the process of and g and locally create motions in new directions. Because controlling several object degrees-of-freedom with fewer the system is not symmetric, however, local controllability robot actuators. Underactuated manipulation takes ad- is sacri ced in the reduction to a single actuator. Instead, vantage of nonholonomy, allowing us to construct dextrous if f can be shown to satisfy a recurrence property [10], the robot systems with only a few actuators. An example of system may still be globally controllable. We are interested underactuated manipulation is manipulation by rolling. By in motion planning and control for these types of dynamic controlling just two or three angular velocity variables de- single-input robot systems which are globally controllable scribing the motion of one smooth body rolling over an- but not locally controllable. In this paper we study a particular dynamic single-input other, it is often possible to fully control the system on the ve-dimensional space describing the relative con gu- rolling system: a spherical ball rolling in an ellipsoidal rations of the bodies in contact. Proof that this is possible, bowl. The principal axes of the bowl are unequal, and and a manipulator built on this principle, is described in the bowl can be translated in a single direction not aligned with any of the principle axes (Figure 1). Because of this, [8], [2] and [11]. These works assume direct control over at least two of the single control variable (acceleration of the bowl) can be the rolling angular velocities. Thus the systems are drift- coupled to di erent directions of motion of the ball relative free with at least two control vector elds, i.e., of the form to the bowl. We show that the state of the ball relative to the bowl can z_ = g1 (z )u1 + g2 (z )u2 ; be controlled using the single control input. More specifit is possible to control the equilibrium orientation where z is the con guration of the system, gi (z ) are con- ically, of the ball (when it rests at the bottom of the bowl) on trol vector elds, and ui are controls, or speeds along the its three-dimensional space SO(3). In our vector elds. The two control vector elds allow us to take previous work (see [3]),con guration we have shown using an ideal repeated Lie brackets, creating other motions that the sys- conservative rolling analysis. Here we this will this astem can follow approximately (such as the parallel parking sumption to allow for dissipation and showrelax that can motion for cars), until we have locally created motion in control the equilibrium orientation of the ball on thewethree every possible direction (the Lie algebra rank condition is dimensional manifolds SO(3) using a single input. We have satis ed). Because the system is symmetric, local control- built an experimental setup to test our ideas. In our exlability follows. periments, we have found an empirical mapping between In this paper we are interested in dynamic rolling systems motions of the bowl and re-orientations of the ball, and with a single input we use that empirical mapping in a path planner to nd motion plans for arbitrarily re-orienting the ball. Experiz_ = f (z ) + g(z )u; mental results are presented. where z is the state of one body relative to another and In Section II we will discuss the results related to conf (z ) is a drift vector eld. In this case, we have reduced trollability of the rolling system. Section III describes the the number of actuators to the bare minimum. If the single- motion planning algorithm. The experimental setup and input system were drift-free, the system would be con ned the numerical results for the motion planner have been de-
scribed in Section IV. Finally we conclude in Section V.
3
II. Controllability of the Rolling System
A. Controllability on SO(3) with Dissipation
Here our goal is only to control the equilibrium orientation of the sphere. Now we consider a rolling system with energy dissipation and specialize the controllability analysis to equilibrium con gurations on the Lie group SO(3). Let U be a nite-dimensional space of parameterized trajectories of the bowl such that the bowl begins and ends at rest with zero net motion. De ne the mapping h : U ! SO(3) yielding the net rotation of the ball after the bowl undergoes a motion u 2 U and the ball again settles to equilibrium.1 Let H = h(U ) = fh(u)ju 2 U g. We wish to investigate the set of reachable orientations from an initial con guration g 2 SO(3) by repeated rotations by elements h 2 H , i.e., con gurations of the form gh1 h2 : : : hn for any positive integer n and any hi 2 H . Assume H is connected and has constant dimension. Because H is invariant on SO(3), it suces to consider the reachable set from the identity g = e. Let R(H ) denote the set of reachable orientations, and R(H ) denote the closure of the reachable set. Theorem 1: If the elements of H are not all rotations about the same xed axis, then R(H ) = SO(3). Proof: If dim(H ) 1, then R(H ) is a Lie subgroup of SO(3) of dimension greater than or equal to one. The only compact Lie groups of dimension less than three are the torus T 2, which is not a subgroup of SO(3), and the one-parameter subgroup SO(2) corresponding to rotations about a xed axis. If H 6 SO(2), then R(H ) must be 2 three-dimensional, hence R(H ) = SO(3). Under very mild assumptions, if R(H ) = SO(3), then R(H ) = SO(3). Theorem 1 implies that if dim(H ) is two or three, then R(H ) = SO(3). Instead, we choose U to be a one-dimensional set of trajectories which move the bowl from rest to rest with the nal position of the bowl coincident with the initial con guration. These trajectories yield a connected one-dimensional set H satisfying the conditions of Theorem 1, implying that the system is controllable by rotations in H . In general, at least three motions of the bowl (three multiplications by elements of H ) are necessary to achieve a desired reorientation of the ball.
1 X−Position (cm)
We have shown that our rolling system is globally controllable [3], though the system is nowhere locally controllable. For detailed results on dynamics and controllability of the rolling system, see [3]. All our previous results (as described in [3]) are for idealized rolling systems without dissipation.
2
0
−1
−2
−3
0
0.2
0.4
0.6
0.8
1 Time (sec)
1.2
1.4
1.6
1.8
2
Fig. 2. Di erent input trajectories (for di erent values of j ).
III. Motion Planning
Based on the result of section II-A, our approach to motion planning is to choose a one-dimensional family of bowl trajectories, empirically nd the corresponding onedimensional reachable set H of ball reorientations, then use this set H in a randomized motion planner to nd a sequence of bowl motions to accomplish a desired ball reorientation. In this section we describe the family of bowl trajectories, the method for interpolating empirical data points to arrive at the curve H , and the motion planning algorithm based on Rapidly Exploring Random Trees (RRT) [7]. A. Input Trajectory of Bowl The trajectory of the bowl is similar in nature to the trajectory described in [9]. Here the trajectory is a fth order polynomial curve with the following constraints: 1. The position at the start of the motion (and also at the end of the motion) is zero. 2. The velocity at the start of the motion (and also at the end of the motion) is zero. We chose the following motion pro le for the bowl:
X (t) = j (t ? T ) + k(t ? T )3 + l(t ? T )5
(1)
where j , k and l are constants, 2T is the time period for the entire motion and t is the current time. The constants j , k and l depend on each other based on the end-point constraints described above. Due to these constraints on position and velocity, we have only one independent parameter which governs the trajectory of the bowl (for a xed time period of motion). If we consider j as our independent input variable (the velocity at time t = T ), then based on the above constraints, 1 Note that the mapping h is invariant to the ball's initial orientation since the ball is perfectly symmetric. k and l are given by k = ? T2j2 and l = Tj4 .
q0
q1
q n-1
qn
Fig. 3. Spherical linear interpolation (slerp) for quaternions.
techniques for constructing di erent types of quaternion curves. The curve tting/interpolation techniques for different curves (Bezier, B-Spline, etc) which are de ned on Euclidean space
Abstract | Underactuated manipulation is the process of controlling several object degrees-of-freedom with fewer robot actuators. Underactuated manipulation allows us to build dextrous robots with only a few actuators. In this paper we explore the possibility of useful dynamic manipulation with only a single actuator. Our case study is a ball rolling in an asymmetrical bowl which can be accelerated along one linear degree-of-freedom. For a real physical system we show that it is possible to control the equilibrium orientation of the ball on the three-dimensional manifold SO(3) using the single input. We have built an experimenFig. 1. The rolling system tal demonstration of this system and we have constructed a motion planner to nd a sequence of motions of the bowl to accomplish a desired reorientation. to an integral curve of the control vector eld and would be of little interest. Because of the existence of a drift vector I. Introduction eld, however, it is possible to take Lie brackets between f Underactuated robotic manipulation is the process of and g and locally create motions in new directions. Because controlling several object degrees-of-freedom with fewer the system is not symmetric, however, local controllability robot actuators. Underactuated manipulation takes ad- is sacri ced in the reduction to a single actuator. Instead, vantage of nonholonomy, allowing us to construct dextrous if f can be shown to satisfy a recurrence property [10], the robot systems with only a few actuators. An example of system may still be globally controllable. We are interested underactuated manipulation is manipulation by rolling. By in motion planning and control for these types of dynamic controlling just two or three angular velocity variables de- single-input robot systems which are globally controllable scribing the motion of one smooth body rolling over an- but not locally controllable. In this paper we study a particular dynamic single-input other, it is often possible to fully control the system on the ve-dimensional space describing the relative con gu- rolling system: a spherical ball rolling in an ellipsoidal rations of the bodies in contact. Proof that this is possible, bowl. The principal axes of the bowl are unequal, and and a manipulator built on this principle, is described in the bowl can be translated in a single direction not aligned with any of the principle axes (Figure 1). Because of this, [8], [2] and [11]. These works assume direct control over at least two of the single control variable (acceleration of the bowl) can be the rolling angular velocities. Thus the systems are drift- coupled to di erent directions of motion of the ball relative free with at least two control vector elds, i.e., of the form to the bowl. We show that the state of the ball relative to the bowl can z_ = g1 (z )u1 + g2 (z )u2 ; be controlled using the single control input. More specifit is possible to control the equilibrium orientation where z is the con guration of the system, gi (z ) are con- ically, of the ball (when it rests at the bottom of the bowl) on trol vector elds, and ui are controls, or speeds along the its three-dimensional space SO(3). In our vector elds. The two control vector elds allow us to take previous work (see [3]),con guration we have shown using an ideal repeated Lie brackets, creating other motions that the sys- conservative rolling analysis. Here we this will this astem can follow approximately (such as the parallel parking sumption to allow for dissipation and showrelax that can motion for cars), until we have locally created motion in control the equilibrium orientation of the ball on thewethree every possible direction (the Lie algebra rank condition is dimensional manifolds SO(3) using a single input. We have satis ed). Because the system is symmetric, local control- built an experimental setup to test our ideas. In our exlability follows. periments, we have found an empirical mapping between In this paper we are interested in dynamic rolling systems motions of the bowl and re-orientations of the ball, and with a single input we use that empirical mapping in a path planner to nd motion plans for arbitrarily re-orienting the ball. Experiz_ = f (z ) + g(z )u; mental results are presented. where z is the state of one body relative to another and In Section II we will discuss the results related to conf (z ) is a drift vector eld. In this case, we have reduced trollability of the rolling system. Section III describes the the number of actuators to the bare minimum. If the single- motion planning algorithm. The experimental setup and input system were drift-free, the system would be con ned the numerical results for the motion planner have been de-
scribed in Section IV. Finally we conclude in Section V.
3
II. Controllability of the Rolling System
A. Controllability on SO(3) with Dissipation
Here our goal is only to control the equilibrium orientation of the sphere. Now we consider a rolling system with energy dissipation and specialize the controllability analysis to equilibrium con gurations on the Lie group SO(3). Let U be a nite-dimensional space of parameterized trajectories of the bowl such that the bowl begins and ends at rest with zero net motion. De ne the mapping h : U ! SO(3) yielding the net rotation of the ball after the bowl undergoes a motion u 2 U and the ball again settles to equilibrium.1 Let H = h(U ) = fh(u)ju 2 U g. We wish to investigate the set of reachable orientations from an initial con guration g 2 SO(3) by repeated rotations by elements h 2 H , i.e., con gurations of the form gh1 h2 : : : hn for any positive integer n and any hi 2 H . Assume H is connected and has constant dimension. Because H is invariant on SO(3), it suces to consider the reachable set from the identity g = e. Let R(H ) denote the set of reachable orientations, and R(H ) denote the closure of the reachable set. Theorem 1: If the elements of H are not all rotations about the same xed axis, then R(H ) = SO(3). Proof: If dim(H ) 1, then R(H ) is a Lie subgroup of SO(3) of dimension greater than or equal to one. The only compact Lie groups of dimension less than three are the torus T 2, which is not a subgroup of SO(3), and the one-parameter subgroup SO(2) corresponding to rotations about a xed axis. If H 6 SO(2), then R(H ) must be 2 three-dimensional, hence R(H ) = SO(3). Under very mild assumptions, if R(H ) = SO(3), then R(H ) = SO(3). Theorem 1 implies that if dim(H ) is two or three, then R(H ) = SO(3). Instead, we choose U to be a one-dimensional set of trajectories which move the bowl from rest to rest with the nal position of the bowl coincident with the initial con guration. These trajectories yield a connected one-dimensional set H satisfying the conditions of Theorem 1, implying that the system is controllable by rotations in H . In general, at least three motions of the bowl (three multiplications by elements of H ) are necessary to achieve a desired reorientation of the ball.
1 X−Position (cm)
We have shown that our rolling system is globally controllable [3], though the system is nowhere locally controllable. For detailed results on dynamics and controllability of the rolling system, see [3]. All our previous results (as described in [3]) are for idealized rolling systems without dissipation.
2
0
−1
−2
−3
0
0.2
0.4
0.6
0.8
1 Time (sec)
1.2
1.4
1.6
1.8
2
Fig. 2. Di erent input trajectories (for di erent values of j ).
III. Motion Planning
Based on the result of section II-A, our approach to motion planning is to choose a one-dimensional family of bowl trajectories, empirically nd the corresponding onedimensional reachable set H of ball reorientations, then use this set H in a randomized motion planner to nd a sequence of bowl motions to accomplish a desired ball reorientation. In this section we describe the family of bowl trajectories, the method for interpolating empirical data points to arrive at the curve H , and the motion planning algorithm based on Rapidly Exploring Random Trees (RRT) [7]. A. Input Trajectory of Bowl The trajectory of the bowl is similar in nature to the trajectory described in [9]. Here the trajectory is a fth order polynomial curve with the following constraints: 1. The position at the start of the motion (and also at the end of the motion) is zero. 2. The velocity at the start of the motion (and also at the end of the motion) is zero. We chose the following motion pro le for the bowl:
X (t) = j (t ? T ) + k(t ? T )3 + l(t ? T )5
(1)
where j , k and l are constants, 2T is the time period for the entire motion and t is the current time. The constants j , k and l depend on each other based on the end-point constraints described above. Due to these constraints on position and velocity, we have only one independent parameter which governs the trajectory of the bowl (for a xed time period of motion). If we consider j as our independent input variable (the velocity at time t = T ), then based on the above constraints, 1 Note that the mapping h is invariant to the ball's initial orientation since the ball is perfectly symmetric. k and l are given by k = ? T2j2 and l = Tj4 .
q0
q1
q n-1
qn
Fig. 3. Spherical linear interpolation (slerp) for quaternions.
techniques for constructing di erent types of quaternion curves. The curve tting/interpolation techniques for different curves (Bezier, B-Spline, etc) which are de ned on Euclidean space
