Jacobian motion and its derivatives

Wolfram Stadler, Peter Ederhard

Mechatronics, No. 11, 2001

Summary

                  The paper proposes an innovative and geometrically appealing derivation of the differential motion of an end effector and the corresponding Jacobian matrix.

The Jacobian matrix and its determinant have receive deep studies in the past, strating from Carl Gustav Jacob Jacobi’s memoir “De determinantibus functionalibus” (1841), when it was first introduced. The paper discusses not on the already well-studied properties of this matrix, but on a fast and less time consuming way to obtain it.

The authors define, based on the Denavit-Hartenber representation, the hand-matrix H_0n the relative rigid motion matrices H_ij, the r_nPn, which locates an arbitrary point P_n in the nth frame and r_0pPn which denotes the absolute position with respect to the fixed base frame. Different kinematic approaches can be taken, notable th vector approach is the most visual and physically appealing and directly correlated with it there is the matric representation, while the authors introduce the use of homogeneous coordinates with the last viewpoint.

The Matrix representation, which follows the same procedure as the vector representation, finally demonstrates the following relation: du_on=J(ξ)dξ, where J(ξ) is the Jacobian matrix (Jacobian instead is the term given to the determinant of the Jacobian matrix in case it is a square matrix), u_on, denoting the six degrees of freedom of a rigid motion, collectively representing ₀dc_0n, _ndc_0n, _nr_nPn, _nω_0ndt, vectors which are related to differential displacement dr_0Pn= dc_0n+dκ_0n × r_nPn . dξ is the column matrix of differentials of the exes variables of an r-axis manipulator.

·       Model

The introduction of homogeneous coordinates is scaled imbedding of expression in Rⁿ into Rⁿ⁺¹, y_n+1≠0 is defined as the scale factor (x_n=y_n/y_n+1 , where (x₁,…,x_n) ∈ Rⁿ and (y₁,…,y_n,y_n+1) ∈ Rⁿ⁺¹. The rigid body motion is then defined with ₀r_0Pn=₀c_0n+A_0nnr_nPn and has the homogeneous representation as ₀r_0Pn= ₀H_0nr_nPn . Finally the differential homogeneous rigid motion of the hand frame is: ₀dr_0Pn=₀H_0nndH_0nnr_nPn . The use of scaling factor y₄=1 in the three dimensional kinematic description in R⁴ allows immediate identification with physical quantities. Differential homogeneous rigid velocity can be obtained through derivation of the position vector or by applying same rule for calculating the differential homogenous rigid position shown in page 569.

Comparison with traditional methods (Paul and Whitney) shown before are then done to prove correctness, the model appears to show correctness and be consistent with their calculations.

Key Concepts

Jacobi Matrix 

Key Results

The authors used the method introduces for the AdeptOne Robot, a Scara configured robot used as a benchmark for computational procedures.

The final Jacobian matrix appears to be straightforward but intensive computation, therefore the authors should make three different computer assisted symbolical and semi-symbolical approaches, the first being implement in Neweul, a program for symbolic generation of the kinematic and kinetic relations for large multi-body systems, the second obtained through Maple, a program under development in conjunction studies between University of Waterloo (Canada) and ETH (Switzerland) and the third being the automatic differentiation (AD) to generate the Jacobian matrix (a semi-numerical/semi-symbolic technique for the computation of derivate).

The AD generates codes for the computation of gradients, they are computer codes not intended for human use.

Two are the basic modes use in automatic differentiation: forward mode and the reverse mode. The former is a gradient vector assigned to each intermediate node x_i, the latter being a scalar derivative of x_i assigned to each node. Derivatives are simply calculated for active nodes for faster computation, an active node is the one with values depending on independent variables, the hand-matrix can be derived in pieces for faster computation.