Robotics I

[Maple Metafile]

Inverse kinematics problem : Given a robot arm with six revolute joints, a typical problem is finding the values of the different joint angles that place the tip or hand at some desired position and orientation.

"At the core of both the analysis of existing machines and the design of new ones is the solution of sets of nonlinear algebraic equations with parameters obtained form either the dimensions of the machine or the designed set of output positions....the designer is less often interested in a particular solution than in ranges of solutions and their relation to the prescribed features of the design..." McCarthy, J.M.: Introduction to: “Kinematics of Robot Manipulators", McCarthy, J.M. editor, MIT press, 1987, and International Journal of Robotics Research, vol.5, no. 2, summer 1986.


[Maple Metafile]

[Maple Metafile]

Determining equation in c_2 s_2

[Maple Metafile]

Now this equation can be rewritten as

[Maple Metafile]

It decomposes (Kovacs, Hommel, z.Gathen, Weiss, Kluners,R-Gutierrez)

Therefore, since f=g(h), solving f(s,c)=0 is reduced to

solving g(x)=0

and for each root “r” of this equation

solving h(s,c)=r

This degree-two sine-cosine equation can be solved by a sequence of one degree-two standard univariate equation and two degree-one sine-cosine equation.

[Maple Metafile]

[Maple Metafile]


The above condition was taken by Smith-Lipkin (1991, Advances in Robot Kinematics, Springer) as the definition of 3R degenerate robots

[Maple Metafile]

(ie. those having a parallel line in the pencil of conics passing through the

four solutions to the corresponding second degree sine-cosine determining equation f(s,c)).

Modulo c^2+s^2-1, the equation f(s,c)=0 factors as


ie. f(s,c)=(x+d)*(x+d’)o(ac+bs) mod. c^2+s^2-1, therefore, it decomposes.

Geometrically this imposes some restrictions to the robots such as

-first two axes intersecting
-first two axes parallel
-second two axes intersecting
-second two axes parallel
-first and third axes intersecting
-first two axes Bennet and d2=0
-first two axes perpendicular d2=0 and d3=-42cos(a3)

• 6R, Stewart Platform (Roth, Mourrain, Merlet, Lazard, Buchberger,

R-González López)

B. Roth, from Stanford.

Inverse kinematics of the 6R robot: 1969-1989.

Seemingly, all boundary conditions for this case were well conditioned:

Important problem,
Engineering school,
Well-developed region

Relevant mathematical consequences: many papers on IEEE Robotics & Automation

Relevant industrial consequences (up to now):