Handbook of Robotics Chapter 1: Kinematics Ken Waldron Jim Schmiedeler Department of Mechanical Engineering Department of Mechanical Engineering Stanford University The Ohio State University Stanford, CA 94305, USA Columbus, OH 43210, USA September 15, 2005 Contents 1 Kinematics 1 1.1 Introduction . 1 1.2 Position and Orientation Representation . 2 1.2.1 Representation of Position . 2 1.2.2 Representation of Orientation . 2 Rotation Matrices . 2 Euler Angles . 3 Fixed Angles . 3 Angle-Axis . 4 Quaternions . 4 1.2.3 Homogeneous Transformations . 5 1.2.4 Screw Transformations . 5 1.2.5 Pluc¨ ker Coordinates . 6 1.3 Joint Kinematics . 6 1.3.1 Revolute . 7 1.3.2 Prismatic . 7 1.3.3 Helical . 8 1.3.4 Cylindrical . 8 1.3.5 Spherical . 8 1.3.6 Planar . 9 1.3.7 Universal . 9 1.3.8 Rolling Contact . 9 1.3.9 Holonomic and Non-Holonomic Constraints . 9 1.3.10 Generalized Coordinates . 9 1.4 Workspace . 9 1.5 Geometric Representation . 10 1.6 Forward Kinematics . 12 1.7 Inverse Kinematics . 12 1.7.1 Closed-Form Solutions . 13 Algebraic Methods . 13 Geometric Methods . 13 1.7.2 Numerical Methods . 14 Symbolic Elimination Methods . 14 Continuation Methods . 14 Iterative Methods . 14 1.8 Forward Instantaneous Kinematics . 14 1.8.1 Jacobian . 15 1.9 Inverse Instantaneous Kinematics . 15 i CONTENTS ii 1.9.1 Inverse Jacobian . 15 1.10 Static Wrench Transmission . 16 1.11 Conclusions and Further Reading . 16 List of Figures 1.1 place holder . 11 1.2 Example Six-Degree-of-Freedom Serial Chain Manipulator. 11 iii List of Tables 1.1 Equivalent rotation matrices for various representations of orientation. 3 1.2 Conversions from a rotation matrix to various representations of orientation. 4 1.3 Conversions from angle-axis to unit quaternion representations of orientation and vice versa. 4 1.4 Conversions from a screw transformation to a homogeneous transformation and vice versa. 6 1.5 Tabulated Geometric Parameters for Example Serial Chain Manipulator. 11 1.6 Forward Kinematics of the Example Serial Chain Manipulator in Figure 1.2. 12 1.7 Inverse Position Kinematics of the Articulated Arm Within the Example Serial Chain Manipulator in Figure 1.2. 13 1.8 Inverse Orientation Kinematics of the Spherical Wrist Within the Example Serial Chain Manipulator in Figure 1.2. 13 1.9 Forward Kinematics of the Example Serial Chain Manipulator in Figure 1.2. 15 iv Chapter 1 Kinematics Kinematics is the geometry of motion, which is to say from the specified values of the manipulator’s joint vari- geometry with the addition of the dimension of time. ables. Inverse kinematics is the determination of the It pertains to the position, velocity, acceleration, and all values of the manipulator’s joint variables required to higher order derivatives of the position of a body in space locate the end-effector in space with a specified position without regard to the forces that cause the motion of the and orientation. The problems of forward and inverse body. Since robotic mechanisms are by their very essence instantaneous kinematics are analogous, but address designed for motion, kinematics is the most fundamental velocities rather than positions. While kinematics does aspect of robot design, analysis, control, and simulation. not consider the forces that generate motion, kinematic Unless explicitly stated otherwise, the kinematic de- velocity analysis is closely related to static wrench analy- scription of robotic mechanisms typically employs a num- sis. The Jacobian that maps the joint velocities to the ber of idealizations. The links that compose the robotic end-effector velocity also maps the end-effector’s trans- mechanism are assumed to be perfectly rigid bodies hav- mitted static wrench to the joint forces/torques that ing surfaces that are geometrically perfect in both posi- produce the wrench. tion and shape. Accordingly, these rigid bodies are con- The goal of this chapter is to provide the reader with nected together at joints where their idealized surfaces an overview of different approaches to representing the are in complete contact without any clearance between position and orientation of a body in space and algo- them. The respective geometries of these surfaces in con- rithms to use these approaches in solving the problems tact determine the freedom of motion between the two of forward kinematics, inverse kinematics, and static links, or the joint kinematics. In an actual robotic wrench transmission for robotic manipulators. mechanism, these joints will have some physical limits beyond which motion is prohibited. The workspace of 1.1 Introduction a robotic manipulator is determined by considering the combined limits and freedom of motion of all of the joints It is necessary to treat problems of the description and within the mechanism. displacement of systems of rigid bodies since robotic A common task for a robotic manipulator is to locate mechanisms are systems of rigid bodies connected by its end-effector in a specific position and with a specific kinematic joints. Among the many possible topologies orientation within its workspace. Therefore, it is critical in which systems of bodies can be connected, two are of to have a representation of the position and orientation of particular importance in robotics: serial chains and fully a body, such as an end-effector, in space. While there are parallel mechanisms. A serial chain is a system of rigid in principle an infinite number of ways to describe the po- bodies in which each member is connected to two others, sition and orientation of a body in space, this chapter will except for the first and last members that are each con- provide an overview of the representations that are par- nected to only one other member. A fully parallel mech- ticularly convenient for the kinematic analysis of robotic anism is one in which there are two members that are mechanisms. Regardless of the selected representation, connected together by multiple joints. In practice, each two fundamental kinematic problems arise for robotic “joint” is often itself a serial chain. This chapter focuses manipulators. Forward kinematics is the determina- almost exclusively on serial chains. Parallel mechanisms tion of the position and orientation of the end-effector are dealt with in more detail in Chapter 13 Kinematic 1 CHAPTER 1. KINEMATICS 2 j Structure and Analysis. the projections of the vector pi onto the corresponding Obviously, the draft of this chapter in its current form axes. The vector components could also be expressed is incomplete. It is missing figures and tables. It is short as the spherical or cylindrical coordinates of Oi in the j on references in many places. The later sections are par- frame. Such representations have advantages for analysis ticularly inadequate in their present form. This Intro- of robotic mechanisms including spherical and cylindrical duction section itself is essentially yet to be written. joints. A translation is a displacement in which no point in the rigid body remains in its initial position and all straight 1.2 Position and Orientation lines in the rigid body remain parallel to their initial ori- Representation entations. The translation of a body in space can be represented by the combination of its position prior to Spatial, rigid body kinematics can be viewed as a com- the translation and its position following the translation. parative study of different ways of representing the po- Conversely, the position of a body can be represented as a sition and orientation of a body in space. Translations translation that takes the body from a position in which and rotations, referred to in combination as rigid body the coordinate frame fixed to the body coincides with the displacements, are also expressed with these representa- fixed coordinate frame to the current position in which tions. No one approach is optimal for all purposes, but the two fames are not coincident. Thus, any representa- the advantages of each can be leveraged appropriately to tion of position can be used to create a representation of facilitate the solution of different problems. displacement, and vice-versa. The minimum number of coordinates required to lo- cate a body in Euclidean space is six: three for position 1.2.2 Representation of Orientation and three for orientation. Many representations of spa- tial position and orientation employ sets with superabun- There is significantly greater breadth in the representa- dant coordinates in which auxiliary relationships exist tion of orientation than in that of position. This section among the coordinates. The number of independent aux- does not include an exhaustive summary, but focuses on iliary relationships is the difference between the number the representations most commonly applied to robotic of coordinates in the set and six. manipulators. This chapter and those that follow it make frequent A rotation is a displacement in which at least one point use of “coordinate reference frames” or simply “frames”. of the rigid body remains in its initial position and not A coordinate reference frame i consists of an origin, de- all lines in the body remain parallel to their initial ori- noted Oi, and a triad of mutually orthogonal basis vec- entations. The points and lines addressed here and in tors, denoted [xˆi yˆi zˆi], that are all fixed within a par- the definition of translation above are not necessarily ticular body. The position and orientation of a body contained within the boundaries of the finite rigid body. in space will always be expressed relative to some other Rather, any point or line in space can be taken to be body, so they can be expressed as the position and orien- rigidly fixed in a body. For example, a body in a cir- tation of one coordinate frame relative to another.