Static Force Analysis:Another Role for the Jacobian

Portions abstracted from H. Asada and J.-J. E. Slotine, “Robot Analysis and Control,” Wiley, 1986

©2018 Max Donath

Static Force Analysis: Balance of forces and moments

Consider forces and moments acting on link i.

Arm links are stationary.

Exerted on the end effector by an external object

Special Cases

Equivalent Joint Torques:

Actuator I exerts scalar torque or force tI between link I-1 and link I in the direction of joint axis I-1. Find the relationship between tI and F˚I-1 or N ˚I-1, I

Moment N ˚I-1, I and other components of F ˚I-1, I are supported by the joint mechanism internal force and moment

All the other components are supported by the joint structure.

Joint torque vector, N x 1

Endpoint force and moment, 6 x1

Definition:

Consider the virtual work done by the N actuators and the environment on the arm linkage

Theorem:

The joint torques t required for a manipulator to exert a given force and moment F at its end point are given by

where friction in the joint mechanisms are ignored and gravity forces are not included. The matrix J is the (6 x N) Jacobian matrix associated with the differential relationship between joint displacements DQ and end point displacements DP:

Proof

Principle of Virtual Work

Let dQ and dP be virtual displacements of the joints and endpoint, respectively.

According to the principle of virtual work, dWork must vanish for arbitrary geometrically admissible displacements, dQ and dP , when the arm linkage is in equilibrium

The joint torques t that balance with external force -F are called the equivalent joint torques.

To resolve the endpoint force to joint torques, one does not need to compute the inverse of the Jacobian matrix.

Example 1: Find the equivalent joint torques to the endpoint force F

For derivation of above J, see Example 5.2.1, Textbook (1989)orExample 4.5, Textbook (2006)

Force relationships across coordinate frames:

If the virtual displacements, δp in a U-V-W coordinate frame are related to

the virtual displacements, δq in an X-Y-Z coordinate frameby

δp = J δq

Then the forces, Q in the X-Y-Z coordinate frameare related to

the forces P in the U-V-W coordinate frameby

Q = JT P

Example to follow…

from H. Asada and J.-J. E. Slotine,“Robot Analysis and Control,” Wiley, 1986, pp 84-86

In an analysis similar to the previous one based on theprinciple of virtual work,

one can develop the following relationship

The wrist force/moment sensormeasures forces and momentsFU, FV, Fw and NU, Nv, NW.

Find the force and moment actingon the tool at point T.

Note: U-V-W is aligned with X-Y-Z in this example

€

DP~

=

DUDVDWDφUDφVDφW

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

=

1 0 0 RZ −RY1 −RZ 0 RX

0 1 RY −RX 00 1 0

0 10 0 1

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

DXDYDZDφXDφYDφZ

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

= J~DQ~

€

DφU =DφX , DφV =DφY , DφW =DφZDU =DX + RZDφY −RY DφZDV =DY −RZDφX + RX DφZDW =DZ + RYDφX −RX DφY

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Servo stiffness and endpoint compliance:

Consider a joint drive system with position feedback control.

Find the endpoint compliance assuming that arm links are rigid

1) Equivalent joint torques to the external endpoint force/moment:

2) Joint deflections

3) Endpoint deflections:

Note A) C varies depending on the arm configuration Q

B) The magnitude of ∆P varies depending on the direction of the applied force F.

Manipulator Dynamics

So far only kinematics considered, what about

the manipulator dynamics and its control?

Problems:

A. Multimode structural resonance at each joint. In a serial manipulator each joint is supported by the preceding link.

Resonant frequency for many manipulators is approximately 10 HZ

Must sample and control at above 60 HZ

Result:

Computational time limit for on-line computer control … 16.7 msec

B. Variation of link moments of inertia with manipulator motion and with payload

C. Large gravitational effects and its variation with manipulator orientation

D. Coordination of joint motion

E. Coriolis and centrifugal effects during reasonably fast motion

Possible Result:

Poor path control for unknown or varying loads