Alba Perez Idaho State University, USA

1. Sizing a Serial Chain to Fit a Task Trajectory Using Clifford Algebra Exponentials Alba Perez Idaho State University, USA J. Michael McCarthy University of California, Irvine, USA International Conference on Robotics and Automation ICRA 2005 Barcelona, Spain April 22, 2005

2. Introduction • Kinematic synthesis of serial chains • Chen and Roth (1967), Tsai and Roth (1973), Lee and Mavroidis (2000) • Clifford algebras, dual quaternions, and product of exponentials • Bottema and Roth (1979), Daniilidis (1999), Murray, Li and Sastry (1994) • Solving constraint synthesis problems • Su, McCarthy and Watson (2004), Perez and McCarthy (2004) • Synthetica 2.0: General serial chain solver • Collins, McCarthy, Perez, and Su (2002)

3. Kinematics Equations • Serial chain: A series of joints connected by links • Joints: R, P, T, C, S • The movement of the serial chain is defined by the direction and location of the joint axes, Si=(Si, CiSi), i = 1,…,m. • The kinematics equations of the robot relate the motion of the end-effector to the composition of transformations about each joint axis. • Matrix representation using Denavit-Hartenberg parameters:

4. Relative Kinematics Equations • Choose a set of joint parameters = (q10, . . ., qn0) to define a reference configuration of the chain denoted D( ). • Compute the relative displacements • This transforms the coordinates of the joint axes to the world frame: • The joint transformations occur about the axes in the reference configuration defined in the world frame:

5. Matrix Exponential Form • The transformations [T(Dqi, Ddi, Si)] can be seen as the matrix exponential of a Lie Algebra element [Ji]: • Let Ji be the screw defined by joint axis Si, Ji = (Si, CiSi + mi Si), with mi = Ddi/Dqi being the pitch of the screw. • The transformation can be written as: • The relative kinematics equations, expressed as a product of exponentials, take the form:

6. Clifford Algebra Exponential • We can represent the screw J = (S, C S +  S) as a Clifford algebra element by defining the dual vector S= S + eC S so we have J = (1+e ) S= sxi+syj+szk+(cysz-czsy+sx)ie(czsx-cxsz+sy)je (cxsy-cysx+sz)ke. • The exponential of this screw is given by • We obtain: • A displacement D about an axis S of angle  and slide d is represented in dual quaternions as the exponential:

7. Clifford Algebra Exponential • Representation of the rotation about a revolute joint of axis S and rotation angle : • Matrix formulation (Lie algebra exponential): • expands to: • Dual quaternion formulation (Clifford algebra exponential):

8. Clifford Algebra Kinematics Equations • Clifford algebra exponential: dual quaternion representing a displacement about an axis S, • The Clifford algebra kinematics equations for a relative displacement are

9. Clifford Algebra Synthesis Equations Finite-position synthesis: Dimension a serial chain so that its end-effector reaches a specified set of spatial positions and orientations. • Specify the m goal positions, P1, …, Pm. • Calculate the relative transformations from the first position and express them as elements of the Clifford algebra, P1j, j=2,..,m. • Equate the kinematics equations to each of the goal transformations, We solve these equations to design serial chains.

10. Clifford Algebra Synthesis Equations • Synthesis equations: • Solve the set of nonlinear equations for the dimensions of the robot (S1,…,Sn) and for the values of the joint variables to reach each position, (1j, …, nj), j = 1, …, m. • Each goal position adds 6 independent equations. Each joint axis is defined by 4 (R joint) or 2 (P joint) structural variables. In order to reach m-1 relative positions, we also have n(m-1) values for the joint variables. • For a robot with r revolute joints and t prismatic joints, designed for a set of m goal positions, we have a system of 6(m-1) equations and 4r+2t+(r+t)(m-1) unknowns. • For example:

11. Structure of the Synthesis Equations Consider the 5C Serial Chain • 5 cylindrical joints, 10 degrees of freedom. • Relative kinematics equations for the 5C chain, Clifford algebra exponential form: • where each axis is defined as: Any 5-jointed serial chain can be derived by specializing the 5C expression.

12. Structure of the Synthesis Equations • The equations can be collected as terms in the products of joint variables, • Products of the sines and cosines of the joint angles (32 monomials), • plus the terms containing the slides. Total: 192 monomials, • Write the kinematics equations as the sum of terms, • where the 8-dimensional column vectors Ki contain the structural variables of the joint axes.

13. CCS Chain Example CCS Serial Chain: • 2 cylindrical joints + 1 sperical joint at the wrist • 7 degrees of freedom Kinematics Equations: • Specialize the 5C kinematics equations, • where L is the set of indices of the monomials that do not contain prismatic variable for axes 3, 4 and 5, L={1, 2,…, 96}. • For instance, the monomial corresponding to index 33, M33 = d1/2 s1s2s3s4s5, has coefficient: K33 = {0, 0, 0, 0, s2y*s3y*s4x + s2z*s3z*s4x - s2y*s3x*s4y - s2z*s3x*s4z + s2x*(s3x*s4x + s3y*s4y + s3z*s4z), -(s2x*s3y*s4x) + s2x*s3x*s4y + s2z*s3z*s4y - s2z*s3y*s4z + s2y*(s3x*s4x + s3y*s4y + s3z*s4z), -(s2x*s3z*s4x) - s2y*s3z*s4y + s2x*s3x*s4z + s2y*s3y*s4z + s2z*(s3x*s4x + s3y*s4y + s3z*s4z), -(s2z*s3y*s4x) + s2y*s3z*s4x + s2z*s3x*s4y - s2x*s3z*s4y - s2y*s3x*s4z + s2x*s3y*s4z};

14. CCS Chain Example Design Equations: • For this example, we specify the motion of the revolute joint and the slide of the cylindrical joint S1,. • Counting: 6(n-1) equations, 11 structural variables, 5(n-1) joint variables. Maximum n = 12 task positions. • Equate the kinematics equations to the n = 12 task positions, • Obtain 66 nonlinear equations in 11 structural and 55 joint variables. Solve numerically to obtain one CCS chain.

15. Example 1: CCS Chain

16. Example 2: CCS Chain 61 sec., 1 run

17. Summary • Clifford algebra exponential provide a convenient formulation of the design equations for serial chains. • These design equations can be used to solve for structural parameters for any serial chain, and if desired for values of the joint parameters as well. • The Clifford algebra formulation yields algebraic equations. The number of terms grows rapidly with the number of joints of the robot. • The 2C robot has 8 structural variables, so for 5 positions we obtain 24 equations with 24 unknowns (8 structural, 16 joint values). Each equation has 12 terms. • The 5C robot has 20 structural variables, so for 21 positions we obtain 120 equations in 220 unknowns (20 structural, 200 joint values). Each equation has 192 terms and 100 free parameters. • The Clifford algebra exponential formulation provides a systematic methodology for creating synthesis equations for serial chains, which is related directly to the coordinates of the joints of the robot. • Future research: Application to variable structure robotic systems to integrate structure specification and path planning.