Geometric Design of MechanicallyReachable Surfaces

J. Michael McCarthy and HaiJun SuUniversity of California, Irvine

Mathematics Seminar

California State University, Long Beach

February 27, 2004

Overview

• Robot manipulators and spatial mechanisms as assemblies of serial chains.

• The seven serial chains with reachable surfaces.

• The model problem: determining a sphere through four points.

• Counting the solutions for a set of polynomial equations.

• The design equations for the seven serial chains.

• The homotopy solution methodology for polynomial equations.

• POLSYS-GLP results.

• Other serial chains and our Synthetica software.

• Conclusions.

Geometric Design

• The kinematic structure of a robot manipulator or spatial mechanism is thenetwork of links and joints that define its movement.

• The primary component of this network is the serial chain of links connected byjoints. One end of the chain is the base, and the other is the end-effector.

Serial Robot PUMA 560 Parallel Robot HEXA

The goal of geometric design is to determine the physical dimensions of a serial chainthat guarantee its end-effector can achieve a specified task.

Reachable Surfaces

PPS chain: PlaneTS chain: Sphere

RPS chain: Hyperboloid

right RRS chain: Circular torus

RRS chain: TorusCS chain: Circular cylinderPRS chain: Elliptic cylinder

The Model Problem

• Consider the problem of determining a sphere that contains four specified points.

• Let R be the radius of the sphere, B=(u, v, w) its center, and P=(X, Y, Z) be a generalpoint on the sphere, then we have

(X-u)2 + (Y-v)2 + (Z-w)2 = R2, or (P-B)⋅(P-B) = R2

Now let Pi=(Xi, Yi, Zi), i=1, 2, 3, 4 be four specified points, so we have

Pi ⋅ Pi - 2 Pi ⋅ B + B ⋅ B = R2, i=1, 2, 3, 4

Subtract the first equation from the remaining to cancel the terms B ⋅ B and R2,

S: (Pi+1 ⋅ Pi+1 - Pi ⋅ Pi) - 2 (Pi+1 - Pi) ⋅ B = 0, i=1, 2, 3.

This is a set of three linear equations in the parameters B=(u, v, w). There canbe at most one solution, which defines the sphere that contains the four points.

Generalizing the Problem

• Rather than specify the points Pi=(Xi, Yi, Zi), we now specifyseven spatial displacements [Ti ]=[Ai, di], that define task positionsfor an end-effector.

• We now seek p=(x, y, z), B=(u, v, w) and R, such that Pi = [Ti ]p lie on the sphere:

(Xi-u)2 + (Yi-v)2 + (Zi-w)2 = R2, or ([Ti ]p-B)⋅([Ti ]p-B) = R2, i=1,…, 7

Subtract the first equation from the remaining to cancel the terms B ⋅ B and R2,

S: (Pi+1 ⋅ Pi+1 - Pi ⋅ Pi) - 2 (Pi+1 - Pi) ⋅ B = 0, i=1,…, 6, where Pi = [Ti ]p .

This is a set of six quadratic equations in the parameters B=(u, v, w) and p=(x, y, z).There can be at most 26=64 solutions.Each solution defines a TS chain that guides its end-effector through the task positions [Ti ].

Counting Solutions

• Total degree: A system of n polynomials of degree d1, d2, … , dn in nvariables has at most D = d1d2… dn isolated solutions. (Bezout, 1779)

• Monomial polytope root count (BKK theory): Proof: Bernshtein, 1975Algorithms: Canny, 1991, Gao and Li, 1999.

• Multihomogeneous root count: Morgan & Sommese, 1987

• General Linear Product root count: Verschelde & Haegemans, 1993,Morgan, Sommese, & Wampler, 1995

General Linear Product

• A polynomial system has the same number, dimension, and degree of solutioncomponents for “almost all” values of the coefficients.

• This means we can count solutions (roots) using a polynomial system that has thesame monomial structure though different coefficients.

Let <u, v, w> denote the linear combination “a1 u + a2v +a3w + a4 “ where ai aregeneric coefficients.

Then the quadratic curve Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 can be written as thelinear product <x, y><x, y>=0

Example: Consider the two plane curves:

C1: A1 x2 + B1 xy + D1 x + E1 y + F1 = 0, <x><x, y>|1 = 0

C2: A2 x2 + B2 xy + D2 x + E2 y + F1 = 0, <x><x, y>|2 = 0

How many solutions? 1

1

1 Three.

The TS Serial Chain

Expanding (Pi+1 ⋅ Pi+1 - Pi ⋅ Pi), we find that the quadratic terms cancel and this termactually has the form <u, v, w>, which means the TS chain equations actually havethe form:

Polynomials with this structure have (63) = 20 roots, which means there are at

most 20 TS chains that can reach the seven task positions.

123and so on for (63).

S: <u, v, w><x, y, z>|1 = 0,<u, v, w><x, y, z>|2 = 0,<u, v, w><x, y, z>|3 = 0,<u, v, w><x, y, z>|4 = 0,<u, v, w><x, y, z>|5 = 0,<u, v, w><x, y, z>|6 = 0.

The equations for the TS chain have the monomial structure:

S: <u, v, w><u, v, w> - <u, v, w><x, y, z>|i = 0 , i=1, … , 6

Design Equations

4,194,304

2,097,152

2,097,152

262,144

16,384

64

32

Total

Degree

448,70211-4th12Torus

(RRS)

868,3521-2nd, 10-4th10Circular Torus

(right RRS)

247,9682-3rd, 9-4th10EllipticCylinder (PRS)

9,2169-4th10Hyperboloid

(RPS)

2,1847-4th8Circular

Cylinder (CS)

206-2nd7Sphere (TS)

105-2nd6Plane (PPS)

GLP

Bound

Equations-

degree

Number

Task Positions

Solving the Design Equations

• The goal is to find all of the real solutions to the design equations. They are areall candidates designs.

• Resultant techniques can be used systems with as many as 50 roots, andeigenvalue elimination methods can extend this to as as high as 100 roots.

• Systems of equations with hundreds and thousands of roots require polynomialhomotopy solution methods.

Polynomial Homotopy

• Let P(z) be the system of polynomial design equations, and we seek all thesolutions z to P(z)=0.

• Now let Q(z) be a polynomial system that has the same monomial structure asP(z), which means we require it to have the same GLP structure.

• The NGLP roots of Q(z)=0 are easily computed by solving linear equations.

• Construct the convex combination homotopy H(λ, z) = (1- λ)Q(z) + λ P(z), whereλ ∈ [0, 1) .

• For each root z = aj of Q(z)=0 the homotopy equation H(λ, z) = 0 defines a zerocurve γj, j=1, …, NGLP, which is a connected component of H-1(0).

Each zero curve of H(λ, z) = 0 leads either to a root of P(z)=0 or a root at infinity.

Tracking Zero Curves

• A zero curve can be parameterized by its arc-length s, so it has the form γj =(λ(s), z(s)). We seek the sequence of points yi ≈ (λ(si), z(si)) along γj.

• Along the zero curve γj, we have H(λ(s), z(s))=0, therefore

The matrix [JH] = [Hλ Hz] is the nx(n+1) matrix of partial derivatives.

• Notice that v = (dλ/ds, dz/ds) is tangent to the zero curve, and it is in thenull-space of [JH].

• This allows us to estimate the next point along γj by the formula yi+1 = yi + (si+1 - si)v(si).

This is essentially numerical integration of an ODE and can be solved with efficientpredictor-corrector methods. Furthermore, it is well-adapted for parallel computation.

Number of Solutions

42,615

94,622

18,120

1,024

804

20

10

Number

Roots

POLSYS-GLP

40m/1024cpu448,7024,194,304Torus

(RRS)

POLSYS-GLP

70m/1024cpu868,3522,097,152Circular Torus

(right RRS)

POLSYS-GLP

30m/8cpu247,9682,097,152Elliptic

Cylinder (PRS)

PHC

24 hrs.9,216262,144Hyperboloid

(RPS)

PHC

5+ hrs.2,18416,384Circular

Cylinder (CS)

resultant2064Sphere (TS)

resultant1032Plane (PPS)

comments

GLP

Bound

Total

Degree

SYNTHETICA 2.0

These results are being integrated into computer-aided design softwarein the Robotics and Automation Laboratory at UCI.

Other Spatial Chains

Here we have discussed seven spatial serial chains. The spatialconstrained serial chains can be enumerated:

There are 15 classes with an additional 35 special cases. Includingpermutations there are 191 chains.

Conclusions

• Kinematic synthesis of spatial chains provides the opportunity to invent newdevices for controlled spatial movement.

• The seven serial chains PPS, TS, RPS, PRS, CS, right RRS, and RRS havereachable surfaces that can be shaped so that the end-effector reaches a largenumber of specified task positions.

• The general cases have a remarkably large number of solutions. Yet 90% ofthe paths traced by our homotopy algorithm are a waste of cpu-time.

• More efficient solution procedures are needed to make computer-aided-invention practical.

• In fact, we are beginning to consider very large scale computing that evaluatesthe solutions of the design equations for large number of serial chains.