IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JAN, 2021 1 A Versatile Inverse Kinematics Formulation for Retargeting Motions onto Robots with Kinematic Loops ,1 ,1 ,1 Christian Schumacher∗ , Espen Knoop∗ , Moritz Ba¨cher∗ Abstract—Robots with kinematic loops are known to have superior mechanical performance. However, due to these loops, their modeling and control is challenging, and prevents a more widespread use. In this paper, we describe a versatile Inverse Kinematics (IK) formulation for the retargeting of expressive mo- input motion tions onto mechanical systems with loops. We support the precise control of the position and orientation of several end-effectors, and the Center of Mass (CoM) of slowly walking robots. Our formulation safeguards against a disassembly when IK targets are moved outside the workspace of the robot, and we introduce a regularizer that smoothly circumvents kinematic singularities where velocities go to infinity. With several validation examples IK, retargeting and three physical robots, we demonstrate the versatility and efficacy of our IK on overactuated systems with loops, and for the retargeting of an expressive motion onto a bipedal robot. Index Terms—Kinematics; Parallel Robots; Methods and Tools for Robot System Design physical robot I. INTRODUCTION Fig. 1. Starting from an input motion, in this case an animated walking cycle (top), our pipeline solves the IK problem of retargeting this motion onto robots OBOTS consisting of kinematic loops have several ad- with arbitrary kinematic structures (middle). By including a CoM objective vantages over designs that consist of a fully actuated in the IK, we obtain a stable gait that can be executed on a physical robot kRinematic tree: they have superior stiffness, allow the place- (bottom). Note that a naive motion retargeting, without the CoM objective, would cause the robot to fall over. ment of actuators where there is space, and often exhibit an advantageous payload-to-weight ratio [1]. However, while there are clear mechanical advantages, kinematic trees remain keep the CoM within the support spanned by the contact of the a popular choice in legged robots and related systems due robot with the ground. See Fig. 1 for an illustrative example to the complexity of the design, control, and simulation of of our pipeline. mechanisms containing loops. Our approach combines a constraint-based formulation for For robots with loops, or parallel robots, designs are com- mechanical joints with an objective-based formulation for IK monly made of submechanisms for which, for example, a targets. This means that the returned IK solution is as close as closed-form analytical model exists, and inverse kinematics possible to the specified targets, in a least-squares sense, while can be formulated by interfacing between submodules. How- we guarantee that the mechanical constraints of the robot are ever, this modeling forgoes the true potential of robots with satisfied exactly. Moreover, we safeguard against kinematic loops, because for every type of submechanism a custom singularities at which velocities go to infinity. module has to be written from scratch [1], unnecessarily We compare our IK formulation to an implicit approach restricting the mechanical design of custom robots. where actuated degrees of freedom are solved for directly, In this paper, we propose a versatile inverse kinematics that with a first-order optimality constraint on forward kinematics. enables the kinematic control of general robotic systems with In contrast to our formulation, the mechanical system disas- (1) position and orientation objectives to precisely control the sembles if not all IK targets can be met. Moreover, in our motion of end-effectors or bodies, and (2) a CoM objective to IK formulation, we only need derivatives, and hence building Manuscript received: Oct, 15, 2020; Revised Dec, 22, 2020; Accepted blocks, from a multibody kinematics implementation, making Jan, 05, 2021. it straightforward to add inverse kinematics support to existing This paper was recommended for publication by Editor Lucia Pallottino upon evaluation of the Associate Editor and Reviewers’ comments. This work forward kinematics codes. was supported by Disney Research. With validation examples, we demonstrate that our IK works ∗All authors contributed equally. on mechanical systems containing all common types of active 1All authors are with Disney Research, Stampfenbachstrasse 48, 8006 Zurich, Switzerland. [email protected] and passive joints. With three physical robot examples, we Digital Object Identifier (DOI): see top of this page. demonstrate how (1) a rich dancing motion can be transferred 2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JAN, 2021 over to a 6-DoF pair of legs with a single end-effector (see, e.g., [18]). However, these mixed representations where objective, (2) an expressive walking cycle can be retargeted reduced and maximal coordinates are combined, are very onto a bipedal robot with complex loops, with the help of complex. In contrast, our kinematics and inverse kinematics is our CoM objective, and (3) how we can safeguard against applicable to general mechanical systems that are modeled as velocity singularities when retargeting motions from a digital multibody systems, represented with generalized coordinates. to a physical arm with fewer degrees of freedom. Our IK Analytic IK solutions are also common (see for exam- can hence be understood as a first-order approximation of a ple [19, 20, 21]), however they are laborious, and do not dynamics model, without having to deal with force and torque generalize to arbitrary mechanisms. transfer, or frictional contact. There are relatively few examples in the literature of serial- parallel robots where the parallel elements cannot be treated as submodules in the serial chain. One such robot is the series- II. RELATED WORK parallel biped presented by Gim et al [22]. We believe that There is a large body of related work on inverse kine- one reason why such robots are not more widespread is their matics in robotics, and related fields such as, e.g., computer challenging inverse kinematics and the lack of suitable off- graphics [2]. While most IK formulations target serial or tree- the-shelf IK solvers. like robots, inverse kinematics for serial-parallel hybrid robots, or mechanical systems that contain kinematic loops, are less III. FORWARD KINEMATICS common despite their desirable properties such as increased Before delving into the specifics of our inverse kinematics, accuracy, stiffness, or payload capacity [1]. We target the latter we summarize how kinematics simulations are performed. To category of robots with our IK. motivate and illustrate our representation, we reference the a) Serial Robots: Tree-like robots are usually mod- four-bar linkage example in Fig. 2. eled as reduced or generalized coordinate formulations, and In what follows, we will first introduce the rigid body closed-form analytical [3], learning-based techniques [4], and representation, and discuss constraints the actuators and joints Jacobian-based numerical approaches [5] are common. Tar- impose. We will then discuss how the unknown states can be geting robotics control, Pechev [6] introduced an IK that solved for after the actuators are stepped. does not require a matrix inversion (increased speed), more robustly handles kinematic singularities, and supports multiple end effectors. Jacobian-based formulations were also extended to handle joint limits (see, e.g., [7]), with Newton-based approaches handling them better in general [8]. Like [9], we enable the control of the CoM position of the robot, and like [6], our IK handles multiple targets gracefully. However, (a) Initial configuration (b) Stepped configuration in contrast, our IK supports arbitrary mechanical systems, Fig. 2. Four-bar linkage in its initial (a) and a stepped configuration (b). The containing kinematic loops. linkage is driven by a single actuator (yellow circle) and has three passive b) Parallel Robots: A large body of work has focused joints (black circles). One of the four bars is fixed relative to the world frame (dark grey). on parallel robots, or Parallel Kinematic Machines (PKMs). An extensive discussion is beyond the scope of this paper, and we refer to [10] for a comprehensive overview. For most A. Representing Rigid Bodies studied PKMs, the IK problem is relatively simple [10] and We represent the position of a rigid body in global coordi- is solved by dividing the mechanism into a number of simple nates with its CoM c R3. This simplifies the formulation of ∈ kinematic chains which are then tackled analytically. A central our CoM objective. Moreover, it means that our method could challenges in the study of PKMs is the avoidance of workspace easily be integrated with common dynamics representations, singularities [11]. More recent work on PKMs includes [12], where the use of frames centered at c is often preferable where the kinematic analysis of an overactuated parallel robot because they enable the decoupled treatment of the linear and is considered, and [13], where a formulation is provided angular motions in the Newton-Euler equations of the bodies. which is applicable also to robots with redundant degrees of To represent a body’s orientation, we rely on 4-coordinate freedom. However, these approaches are not readily applicable unit quaternions q, as they do not suffer from singularities. to arbitrary robot topologies as considered here. This comes at the cost of enforcing unit length constraints. c) General Serial-Parallel Robots: To represent gen- Summarily, we represent the i-th component in an assembly eral robots with loops, maximal coordinates are commonly with a 7-coordinate state vector si, consisting of the position used [14]. As a recent survey on serial-parallel hybrid robots and orientation of the body, and stack them for an n-body points out [1], the “kinematic analysis of generic series-parallel assembly to represent its maximal coordinate state with the hybrid robots is an open problem”, let alone inverse kine- state vector s R7n.