PUMA Robot Dynamics

PUMA 560 robot Dynamics:

Main reference: Brian Armstrong, Oussama Khatib & Joel Burdick , “The Explicit Dynamic Model and Inertial Parameters of the PUMA 560 Arm” , Stanford University, Artificial Intelligence Laboratory, IEEE 1986

Reference2: Abdel-Razzak MERHEB, “Nonlinear Control Algorithms applied to 3 DOF PUMA Robot”, METU 2008

M (q).q˙˙ V (q,q˙).q˙  G(q)   (2.17) where, q : nx1 position vector , M (q) : nxn inertia matrix of the manipulator, V (q,q˙) : nx1 vector of Centrifugal and Coriolis terms G(q) : nx1 vector of gravity terms  : nx1 vector of torques

By writing the velocity dependent term V (q,q˙) in a different form, all the matrices become functions of only the manipulator position; in this case the dynamic equation is called configuration space equation and has the following form:

  M (q).q˙˙  B(q).[q˙.q˙]  C(q).[q˙ 2 ]  G(q) (2.18) where, B(q) : nxn(n-1)/2 matrix of Coriolis torques C(q) : nxn matrix of Centrifugal torques [q˙q˙]: n(n-1)/2x1 vector of joint velocity products given by:

T [q˙1.q˙ 2 ,q˙1.q˙ 3 ,...,q˙1.q˙ n ,q˙ 2 .q˙ 3 ,q˙ 2 .q˙ 4 ,...,q˙ n2 .q˙ n ,q˙ n1.q˙ n ]

2 2 2 2 [q˙ ]: nx1 vector given by: [q˙1 ,q˙ 2 ,....,q˙ n ]

In this context, the configuration space equation presented in [6] will be used. The robot is disassembled and the inertial properties of each link are found. The explicit dynamic model is then obtained with a derivation procedure comprised of several heuristic rules for simulation.

To derive the model of the robot arm, Khatib et al. started by generating the kinetic energy matrix and gravity vector symbolic elements by performing the summation of either Lagrange’s or the Gibbs-Alembert formulation; these elements are then simplified by combining inertia constants that multiply common variable expressions. The Coriolis and centrifugal matrix elements are then calculated in terms of partial derivatives of kinetic energy, and then reduced using four relations that hold on the partial derivatives. Finally, the necessary partial derivatives are formed, and the Coriolis and centrifugal matrices are found. A simplification step is then done by combining the inertia constants that multiply the common variable expressions.

Recall that only three links of PUMA robot are used in this thesis, q4  q5  q6  0 . The configuration space equation is,

  A(q).q˙˙  B(q).[q˙.q˙]  C(q).[q˙ 2 ]  G(q) (2.19) With, Matrix A is a symmetric 6x6 matrix:

a11 a12 a13 0 0 0  a a a 0 0 0   21 22 23 

a31 a32 a33 0 a35 0  A(q)    (2.20)  0 0 0 a44 0 0   0 0 0 0 a 0   55   0 0 0 0 0 a66  Where, a11  I m1  I1  I 3 .CC2  I 7 .SS23 I10 .SC23 I11.SC2  I 21.SS23

 2.I 5 .C2.S23 I12 .C2.C23  I15 .SS23 I16 .C2.S23  I 22 .SC23

a12  I 4 .S2  I 8 .C23 I 9 .C2  I13 .S23 I18 .C23

a13  I 8 .C23  I13 .S23  I18 .C23

a22  I m2  I 2  I 6  2.I 5 .S3  I12 .C2  I15  I16 .S3

a23  I 5 .S3 I 6  I12 .C3 I16 .S3 2.I15

a33  I m3  I 6  2.I15

a35  I15  I17

a44  I m4  I14

a55  I m5  I17

a66  I m6  I 23

a21  a12 , a31  a13 and a32  a23

While matrix B is:

b112 b113 0 b115 0 b123 0 0 0 0 0 0 0 0 0  0 0 b 0 0 b 0 b 0 0 b 0 0 0 0  214 223 225 235   0 0 b314 0 0 0 0 0 0 0 0 0 0 0 0 B(q)    (2. b412 b413 0 b415 0 0 0 0 0 0 0 0 0 0 0  0 0 b 0 0 0 0 0 0 0 0 0 0 0 0  514   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21) Where, b112  2.[I 3 .SC2  I 5 .C223  I 7 .SC23  I12 .S223  I15 .2.SC23  I16 .C223  I 21.SC23 

 I 22 .(1  2.SS23)]  I10 .(1  2.SS23)  I11.(1  2.SS2) b113  2.[I 5 .C2.C23 I 7 .SC23 I12 .C2.S23 I15 .2.SC23 I16 .C2.C23  I 21.SC23

 I 22 .(1 2.SS23)] I10 .(1 2.SS23) b115  2.[SC23  I15 .SC23  I16 .C2.C23  I 22 .CC23]

b123  2.[I 8 .S23 I13 .C23  I18 .S23]

b214  I14 .S23  I19 .S23 2.I 20 .S23.(1 0.5)

b223  2.[I12 .S3 I 5 .C3 I16 .C3]

b225  2.[I16 .C3 I 22 ]

b235  2.[I16 .C3 I 22 ]

b314  2.[I 20 .S23.(1 0.5)] I14 .S23 I19 .S23

b412  b214  [I14 .S23  I19 .S23  2.I 20 .S23.(1  0.5)]

b413  b314  2.[I 20 .S23.(1  0.5)]  I14 .S23  I19 .S23

b415  I 20 .S23 I17 .S23

b514  b415  I 20 .S23 I17 .S23

Matrix C is:

 0 c12 c13 0 0 0 c 0 c 0 0 0  21 23  c31 c32 0 0 0 0 C(q)    (2.22)  0 0 0 0 0 0 c c 0 0 0 0  51 52   0 0 0 0 0 0 Where, c12  I 4 .C2  I 8 .S23  I 9 .S2  I13 .C23 I18 .S23

c13  0.5.b123  I 8 .S23  I13 .C23  I18 .S23 c21  0.5.b112  I 3 .SC2  I 5 .C223 I 7 .SC23  I12 .S223 I15 .2.SC23 I16 .C223 I 21.SC23

 I 22 .(1 2.SS23)  0.5.I10 .(1 2.SS23)  0.5.I11.(1 2.SS2) c23  0.5.b223  I12 .S3 I 5 .C3 I16 .C3

c31  0.5.b113  I 5 .C2.C23  I 7 .SC23  I12 .C2.S23  I15 .2.SC23  I16 .C2.C23  I 21.SC23 

 I 22 .(1  2.SS23)  0.5.I10 .(1  2.SS23) c32  c23  I12 .S3 I 5 .C3 I16 .C3

c51  0.5.b115  SC23 I15 .SC23 I16 .C2.C23  I 22 .CC23

c52  0.5.b225  I16 .C3 I 22

And matrix G is:  0  g   2  g 3  g(q)    (2.23)  0  g   5   0  g 2  g1.C2  g 2 .S23 g 3 .S2  g 4 .C23 g 5 .S23

g 3  g 2 .S23 g 4 .C23 g 5 .S23

g 5  g 5 .S23

Where, Si = sin(θi), Ci = cos(θi), Cij = cos(θi+ θj), Sijk = sin(θi +θj +θk), CCi = cos(θi).cos(θi) and Csi = cos(θi).sin(θi)

Tables 2. and 3. contain the computed values for the constants appearing in the equations of forces of motion, I1  1.43  0.05 I 2  1.75  0.07

I 3  1.38  0.05 I 4  0.69  0.02

I 5  0.372  0.031 I 6  0.333 0.016

I 7  0.298  0.029 I 8  0.134  0.014

I 9  0.0238  0.012 I10  0.0213 0.0022

I11  0.0142  0.0070 I12  0.011 0.0011

I13  0.00379  0.0009 I14  0.00164  0.000070

I15  0.00125  0.0003 I16  0.00124  0.0003

I17  0.000642  0.0003 I18  0.000431 0.00013

I19  0.0003  0.0014 I 20  0.000202  0.0008

I 21  0.0001 0.0006 I 22  0.000058  0.000015

I 23  0.00004  0.00002 I m1  1.14  0.27

I m2  4.71  0.54 I m3  0.827  0.093

I m4  0.2  0.016 I m5  0.179  0.014

I m6  0.193 0.016

Table 2. Inertial Constants ( kg.m 2 )

Table 3, Gravitational Constants (N.m) g1  37.2  0.5 g 2  8.44  0.20

g 3 1.02  0.50 g 4  0.249  0.025

g 5  0.0282  0.0056

Note: tolerance is due to measurement and calculation errors.

2.4. Using PUMA Robot as 3-DOF Robot:

The three degree of freedom PUMA robot has the same configuration space equation general form as its 6-DOF convenient. In this type, the last three joints are blocked so they keep their initial states while the robot is moving. Using the configuration equation of the robot, and by setting the last joints as zero always, we can define a general equation that allows us to use PUMA robot as a 3-DOF robot.

To find the kinematics of the 3-DOF robot, a new D-H coordinate system is established, and a homogenous transformation matrix relating the 3rd coordinate frame to the first coordinate frame is developed. However, the 3-DOF PUMA will have the same kinematics of its 6-DOF convenient with q4 ,q5 and q6 set to zero.

For the configuration space equation of the robot

  A(q).q˙˙  B(q).q˙q˙  C(q).q˙ 2  g(q)

We set q4  q5  q6  0, this yields

T q˙˙  q˙˙1...q˙˙2 ...q˙˙3 ...0...0...0  ,

T q˙q˙   q˙ 1q˙ 2 ...q˙ 1q˙ 3 ...0...0...0...q˙ 2 q˙ 3 ...0...0...0...0...0...0...0...0...0 ,

2 2 2 2 T q˙  q˙ 1 ...q˙ 2 ...q˙ 3 ...0...0...0 , T B(q).q˙q˙  b112 .q˙ 1q˙ 2  b113 .q˙ 1q˙ 3  b123 .q˙ 2 q˙ 3 ...b223 .q˙ 2 q˙ 3 ...0...b412 .q˙ 1q˙ 2  b413 .q˙ 1q˙ 3 ...0...0 and

2 2 2 2 2 2 2 2 2 T C(q).q˙  c12 .q˙ 2  c13 .q˙ 3 ...c21.q˙ 1  c23 .q˙ 3 ...c31.q˙ 1  c32 .q˙ 2 ...0...c51.q˙ 1  c52 .q˙ 2 ...0 The angular acceleration is found as to be q˙˙  A 1 (q).  B(q).q˙q˙  C(q).q˙ 2  g(q)

Now let I    B(q).q˙q˙  C(q).q˙ 2  g(q) q˙˙  A 1 (q).I

2 2 I1  1 b112 .q˙ 1q˙ 2  b113 .q˙ 1q˙ 3  b123 .q˙ 2 q˙ 3  c12 .q˙ 2  c13 .q˙ 3 

2 2 I 2  2 b223 .q˙ 2 q˙ 3  c21.q˙ 1  c23 .q˙ 3  g 2

2 2 I 3  3  c31.q˙ 1  c32 .q˙ 2  g 3

I 4  4 b412 .q˙ 1q˙ 2  b413 .q˙ 1q˙ 3 

2 2 I 5  5  c51 .q˙ 1  c52 .q˙ 2  g 5

I 6  6

These equations tell us that in order to ensure that q˙˙4 ,q˙˙5 and q˙˙6 keep their zero values, it is better to set I 4  I 5  I 6  0 ; so by holding the control torques of the last three joints as

4  b412 .q˙ 1q˙ 2  b413 .q˙ 1q˙ 3 

2 2 5  c51.q˙ 1  c52 .q˙ 2  g 5 and 6  0 , the last three joints are blocked at their initial states. Fig.2.4.1 D-H Coordinate frames of the 3DOF PUMA Robot [6]