Kinematic Predictor for Articulated Mechanisms

• US 20120303318A1
• Filed: 08/06/2012
• Published: 11/29/2012
• Est. Priority Date: 09/28/2005
• Status: Active Grant

• Current Assignee: Wisconsin Alumni Research Foundation (University of Wisconsin System)
• Original Assignee: Wisconsin Alumni Research Foundation (University of Wisconsin System)
• Inventors: Milenkovic, Paul H.
• Granted Patent: US 9,031,699 B2
• Time in Patent Office: Days
• US Class Current: 702/150
• CPC Class Codes: A61B 18/18 by applying electromagnetic radiation, e.g. microwaves radiation therapy A61N5/00
• CPC Class Codes: A61B 18/1815 using microwaves
• CPC Class Codes: A61B 2018/00023 closed, i.e. without wound contact by the fluid

Abstract

A predictor usable for rapid and accurate calculation of joint commands of an articulated mechanism describes relationship between the joints in the form of a differential equation. The predictor solves this differential equation by direct substitution of a power series for each of its variables and the combining of selected sets of coefficients of these power series into linear systems of equations which may be solved to determine power series coefficients to arbitrary order.

Claims

1. A predictor system for an articulated linkage providing a system of links connected by joints including at least one ground link referenced to a global ground reference, the predictor system comprising:

an electronic computer executing a stored program held in non-transitory media to;

(1) store a system of differential equations describing a rate of change in a joint state, each joint as a function of other joint states, the joint states describing positions of the joints and held as joint state variables, where the joints include at least one work joint;

(2) expand the joint state variables of the system of differential equations with power series representations of the joint state variables;

(3) receive an input describing a trajectory of at least one joint state variable of the at least one work joint;

(4) determine values of multiple orders of coefficients of the power series representations of the state variables for joints other than the at least one work joint satisfying a linear relationship among power series coefficients for the state variables; and

(5) output data describing displacement of the joints other than the at least one work joint necessary to produce the input trajectory of the joint state variable of the at least one work joint.

2. The predictor system of claim 1 wherein the values of multiple orders of coefficients of the power series representations of the state variables for joints other than the at least one work joint are determined by solving a linear system of equations in an order-recursive fashion in which lower orders of the coefficients of the power series are computed to provide values for a computation of later orders of the power series.

3. The predictor system of claim 2 wherein the power series representations include power series terms to at least three orders.

4. The predictor system of claim 3 wherein the joints described by the system of differential equations form a loop starting at the ground link and proceeding through one or more joints to a second link to ground.

5. The predictor system of claim 4 wherein the second link is a virtual link that is not part of the articulated linkage.

6. The predictor system of claim 4 wherein the joint states are parameters of joint screws defining an axis and pitch along the axis and the rate of change.

7. The predictor system of claim 6 wherein the input trajectory is expressed as a rate of change of the at least one work joint.

8. The predictor system of claim 6 wherein the output provides a time history of joint screws of the joints other than the at least one work joint.

9. The predictor system of claim 6 wherein the system of differential equations is comprised of elements:

{dot over (T)}_i(t)=T_Ci(t)×

T_i(t)

where;

×

is a Lie product;

i is an index for the joints;

the initial conditions to the differential equation are of the form

T_i=(ω

_i;

r_i×

ω

_i+h_iω

_i) that provides a description of joint axis in Plü

cker coordinates;

10. The predictor system of claim 4 wherein the system of differential equations specifies Denavit-Hartenberg displacement matrices.

11. The predictor system of claim 1 wherein each joint includes an actuator controllably changing a joint position and wherein the outputs are adapted to drive the actuators.

12. The predictor system of claim 11 wherein each joint includes a sensor sensing a joint position and wherein the electronic computer further executes the stored program to receive inputs from the sensors providing joint positions of the joints.

13. The predictor system of claim 1 further including a user input terminal providing an input and a display and wherein the electronic computer further executes the stored program to receive inputs from a user describing joint numbers, joint types and linkage types and to modify template equations according to those inputs of joint numbers, joint types and link types to generate the system of differential equations.

14. A method of predicting motion of an articulated linkage, the articulated linkage providing a system of links connected by joints including at least one ground link referenced to a global ground reference, the method comprising the steps of executing a program stored in non-transitory medium on an electronic computer to:

(1) store a system of differential equations describing a rate of change in a joint state, each joint as a function of other joint states, the joint states describing positions of the joints and held as joint state variables, where the joints include at least one work joint;

(2) expand the joint state variables of the system of differential equations with power series representations of the joint state variables;

(3) receive an input describing a trajectory of at least one joint state variable of the at least one work joint;

(4) determine values of multiple orders of coefficients of the power series representations of the state variables for joints other than the at least one work joint satisfying a linear relationship among power series coefficients for the state variables; and

(5) output data describing displacement of the joints other than the at least one work joint necessary to produce the input trajectory of the joint state variable of the at least one work joint.

15. The method of claim 14 wherein the values of multiple orders of coefficients of the power series representations of the state variables for joints other than the at least one work joint are determined by solving a linear system of equations in an order-recursive fashion in which lower orders of the coefficients of the power series are computed to provide values for a computation of later orders of the power series.

16. The method of claim 15 wherein the power series representations include power series terms to at least three orders.

17. The method of claim 16 wherein the joints described by the system of differential equations form a loop starting at the ground link and proceeding through one or more joints to a second link to ground.

18. The method of claim 16 wherein each joint of the articulated linkage includes an actuator controllably changing a joint position and further including the step of communicating how the output are adapted to drive the actuators.

19. The method of claim 16 wherein each joint further includes a sensor sensing a joint position and further including the step of receiving inputs from the sensors providing joint positions of the joints.