1 Robotic Contact Juggling J. Zachary Woodruff, (Member, IEEE), and Kevin M. Lynch, (Fellow, IEEE) Abstract—We define “robotic contact juggling” to be the purposeful control of the motion of a three-dimensional smooth Start object as it rolls freely on a motion-controlled robot manipulator, or “hand.” While specific examples of robotic contact juggling have been studied before, in this paper we provide the first general formulation and solution method for the case of an arbitrary smooth object in single-point rolling contact on an arbitrary smooth hand. Our formulation splits the problem into four subproblems: (1) deriving the second-order rolling End kinematics; (2) deriving the three-dimensional rolling dynamics; t = 0 t = tf (3) planning rolling motions that satisfy the rolling dynamics; and (4) feedback stabilization of planned rolling trajectories. The Fig. 1. Example of a contact juggling skill known as “the butterfly.” A smooth theoretical results are demonstrated in simulation and experiment object is initially at rest in the palm of the hand, and the motion of the hand using feedback from a high-speed vision system. causes the object to roll to the back of the hand. I. INTRODUCTION simulation and experiment using feedback from a high-speed ONTACT juggling is a form of object manipulation vision system. C where the juggler controls the motion of an object, often a crystal ball, as it rolls on the juggler’s arms, hands, torso, or even shaved head. The manipulation is nonprehensile (no A. Background form- or force-closure grasp) and dynamic, i.e., momentum plays a crucial role. An example is shown in Figure 1. This is When a three-dimensional rigid body (the object) is in a variation of a contact juggling skill called “the butterfly,” and single-point contact with another rigid body (the hand), the robotic implementations of the butterfly have been described configuration of the object relative to the hand is five dimen- in [1]–[3]. The object (typically a ball) is initially at rest on sional: the six degrees of freedom of the object subject to the the palm, and the goal state is rest on the back of the hand. single constraint that the distance to the hand is zero. This The hand is accelerated to cause the object to roll up and over five-dimensional configuration space can be parameterized by the fingers to the other side of the hand. two coordinates uo = (uo; vo) describing the contact location We define “robotic contact juggling” to be the purposeful on the surface of the object, two coordinates uh = (uh; vh) control of the motion of a three-dimensional smooth object describing the contact location on the surface of the hand, as it rolls freely on a motion-controlled robot manipulator, and one coordinate describing the angle of “spin” between or “hand.” Specific examples of robotic contact juggling have frames fixed to each body at the contact point. Collectively been studied before, such as the butterfly example mentioned the contact configuration is written q = (uo; vo; uh; vh; ) above and specific geometries such as a sphere rolling on (Figure 2). a motion-controlled flat plate. This paper extends previous Rolling contact is maintained when there is no relative linear work by providing the first general formulation and solu- velocity at the contact vrel = (vx; vy; vz) = 0 (i.e. no slipping arXiv:2102.10421v1 [cs.RO] 20 Feb 2021 tion method for the case of an arbitrary smooth object in or separation). For rolling bodies modeled with a point contact, single-point rolling contact on an arbitrary smooth hand. no torques are transmitted through the contact, and relative Our formulation splits the problem into four subproblems: spin about the contact normal is allowed. We refer to this (1) deriving the second-order rolling kinematics; (2) deriving as “rolling.” For rolling bodies modeled with a soft contact, the three-dimensional rolling dynamics; (3) planning rolling torques can be transmitted and no relative spin about the motions that satisfy the rolling dynamics and achieve the contact normal is allowed (!rel;z = !z = 0). We refer to desired goal state; and (4) feedback stabilization of planned this as “pure rolling” (or “soft rolling”). rolling trajectories. The theoretical results are demonstrated in This work supported in part by the NSF Graduate Research Fellowship B. Paper Outline Program under Grant DGE-1324585, and in part by the NSF under Grant IIS-1527921. As mentioned above, our approach to contact juggling J. Z. Woodruff ([email protected], corresponding author) and K. M. Lynch ([email protected]) are with the Center for divides the problem into four subproblems: (1) deriving the Robotics and Biosystems (CRB), Northwestern University, Evanston, IL second-order rolling kinematics; (2) deriving the rolling dy- 60208 USA. K. M. Lynch is also affiliated with the Northwestern Institute on namics; (3) planning rolling motions that satisfy the dynamics; Complex Systems, Evanston, IL 60201 USA. Open source code that accompanies this paper can be found at: https: and (4) feedback stabilization of rolling trajectories. An outline //github.com/zackwoodruff/rolling dynamics of each subproblem is given below. 2 1) Rolling Kinematics: First-order kinematics models the vh uo evolution of contact coordinates q between two rigid bodies increasing increasing when the relative contact velocities are directly controlled. y ch The second-order kinematics is a generalization of the first- y order model where the relative accelerations at the contact co nc ch ,ph are controlled. The second-order kinematics is used in our o o { } h { } n { } derivation of the rolling dynamics to describe the evolution of v c o,po ch u o { } x h the contact coordinates during rolling motions. In Section IV increasing ch increasing we describe the second-order kinematics, which include the ψ x work of Sarkar. et al. [4], our corrections to that work [5], co and our new expression for the acceleration constraints that Object Hand enforce pure rolling. 2) Rolling Dynamics: Given the acceleration of the hand Fig. 2. The object and hand are in contact at the origin of frames fcog and and the state of the hand and the object, the rolling dynamic fchg, but they are shown separated for clarity. Two coincident contact frames fpig and fcig for i 2 [o; h] are given for each body at the contact, where equations calculate the acceleration of the object and the fpig is fixed to the object and fcig is fixed in the inertial frame fsg. The contact wrench between the hand and the object. In Section V surfaces of the object and hand are orthogonally parameterized by (uo; vo) and (u ; v ), respectively. At the point of contact, the x - and y -axes of we combine the rolling kinematics with the Newton-Euler h h ci ci the coordinate frames (fcig,fpig) are in the direction of increasing ui (and dynamic equations to derive an expression for the rolling constant vi) and increasing vi (and constant ui), respectively, and the contact dynamics. This formulation enforces rolling contact and calcu- normal nci is in the direction xci ×yci . Rotating frame fchg by about the n -axis of frame fc g aligns the x -axis of frame fc g and the x -axis lates the required contact wrench, which then can be checked co o ch h co of frame fc g. to see if it satisfies normal force constraints (no adhesion) and o friction limits. We validate the simulation of rolling dynamics using the analytical solutions of a sphere rolling on a spinning a generalization of the first-order model where the relative plate. accelerations at the contact are specified. 3) Rolling Motion Planning: In Section VI we use direct Montana derives the first-order contact kinematics for two collocation methods and the rolling dynamics equations to plan 3D objects in contact [6]. His method models the full five- dynamic rolling motions for an object rolling on a manipulator. dimensional configuration space, but it is not easily gener- Given an initial state, we find a set of manipulator controls that alized to second-order kinematics. Harada et al. define the brings the system to the goal state. concept of neighborhood equilibrium to perform quasistatic 4) Feedback Control: In Section VII we demonstrate the regrasps of an object rolling on a flat manipulator using use of a Linear Quadratic Regulator (LQR) feedback controller Montana’s kinematics equations [7]. First- and second-order to stabilize a nominal rolling trajectory. contact equations were derived by Sarkar et al. in [4] and The motion planning and feedback control are validated republished in later works [8], [9]. Errors in the published experimentally in Section VIII. equations for second-order contact kinematics in [4], [8], [9] A video that accompanies this paper can be found in the sup- were corrected in our recent work [5]. Another of our recent plemental media or at: youtu.be/QT55 Q1ePfg. Open-source works covers motion planning and feedback control for first- MATLAB code to derive and simulate open-loop rolling can order-kinematic rolling between two surfaces [10]. be found here: github.com/zackwoodruff/rolling dynamics. Each of [4]–[10] assumes an orthogonal parameterization, as shown in Figure 2. Recent work by Xiao and Ding derives the C. Statement of Contributions second-order kinematics equations for non-orthogonal surface parameterizations [11]. This paper provides the first formulation of the rolling dynamics of a smooth rigid body rolling in point contact on a second, motion-controlled, smooth rigid body. We show that B. Dynamic Rolling the equations can be used for simulation or in optimization- based motion planning considering constraints on contact The evolution of dynamic rolling systems is governed by normal forces and friction limits.