Method and apparatus for creating time-optimal commands for linear systems

• US 20030018400A1
• Filed: 05/21/2002
• Published: 01/23/2003
• Est. Priority Date: 03/05/1998
• Status: Active Grant

• Current Assignee: Timothy D. Tuttle, Warren P. Seering
• Original Assignee: Timothy D. Tuttle, Warren P. Seering
• Inventors: Tuttle, Timothy D., Seering, Warren P.
• Granted Patent: US 6,694,196 B2
• Time in Patent Office: Days
• US Class Current: 700/29
• CPC Class Codes: B25J 9/1605 Simulation of manipulator lay-out, design, modelling of manipulator
• CPC Class Codes: G05B 13/047 the criterion being a time optimal performance criterion
• CPC Class Codes: G05B 2219/35585 Motion command profile
• CPC Class Codes: G05B 2219/39229 Linear parameterization of robot dynamics
• CPC Class Codes: G05B 2219/39358 Time optimal control along path for singular points, having veloctiy constraints
• CPC Class Codes: G05B 2219/43168 Motion profile planning for point to point control

Abstract

A system for and method of determining an input command profile for substantially any dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state. The present invention involves identifying characteristics of the dynamic system, selecting a command profile which defines an input to the dynamic system based on the identified characteristics, wherein the command profile comprises one or more pulses which rise and fall at switch times, imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints. The characteristics may be related to poles and zeros of the dynamic system, and the plurality of constraints may include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a first state to a second state such that the dynamic system remains substantially at the second state.

Claims

1. A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of:

identifying characteristics of the dynamic system;

selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system;

imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times; and

determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.

2. A method according to claim 1, wherein the characteristics identified in the identifying step relate to poles and zeros of the dynamic system.

3. A method according to claim 1, wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a first state to a second state such that the dynamic system remains substantially at the second state.

4. A method according to claim 1, wherein the plurality of constraints include boundary condition constraints, the boundary condition constraints including an initial time of the input and a specified change “

dy”

in the output of the system.

5. A method according to claim 4, wherein the boundary condition constraints are based on the following equations:

t₀=0

6. A method according to claim 1, wherein the plurality of constraints include an actuator limit constraint, the actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator.

7. A method according to claim 1, wherein the determining step comprises determining the switch times based on:

(i) a command profile, u(t), comprising;

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

where “

a_j”

represents an amplitude of pulses in the function, “

t_j”

represents switch times for the pulses, “

c_j”

represents coefficients of exponential terms in a tail of the command profile, “

z_j”

represents zeros in the system, “

n”

represents a total number of switches excluding a first switch, and “

r”

represents a number of zeros in the system,

(ii) a dynamics cancellation constraint comprising;

${\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-s_{t_j}}+{\mathrm{<br><br>}}^{-s_{t_n}}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left[c_j<br><br>t_n^{\left(q_i-1\right)}+\underset{k=1}{\overset{q_i}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left({\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_i-k\right)}\right]\right)<br><br>}_{s=p_i}=0,$

where “

p_i”

represents system poles, “

q_i”

represents a multiplicity of each system pole “

p₁”

, and where β

is defined as follows;

β

₁(k)=1,k=1=0,k≠

1β

_q1(k)=β

<sub>q1−

1</sub>(k)+(k−

1)β

<sub>q1−

1</sub>(k−

1),

(iii) boundary condition constraints comprising;

t₀=0

$\mathrm{dy}=\underset{s\to <br><br>0}{\lim }<br><br>\left(\frac{N<br><br>\left(s\right)}{D<br><br>\left(s\right)}\right)<br><br>\left({\int <br><br>}^{\left(\mathrm{rbp}\right)}<br><br>u<br><br>\left(t\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>t\right)<br><br>\mathrm{LINESUBt}=\mathrm{\infty <br><br>},$

where N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “

rbp”

corresponds to rigid body poles in the system, and

(iv) an actuator limit constraint comprising;

u_min≦

u_tail(t)≦

u_max,

where u_min represents a minimum value of the input, u_tail(t) represents a value of a tail of the input at time “

t”

, and u_max represents a maximum value of the input.

8. A method according to claim 7, wherein the switch times are determined using an optimization algorithm.

9. A method according to claim 1, wherein, in a case that the identifying step identifies that the dynamic system contains poles only and no zeros, (i) the command profile comprises:

10. A method according to claim 9, wherein (i) the dynamics cancellation constraint comprises:

11. A method according to claim 1, further comprising the step of verifying optimality of the input which includes the switch times determined in the determining step.

12. A method according to claim 11, wherein the optimality of the input which includes the switch times determined in the determining step is verified based on Pontryagin'"'"'s Minimum Principle.

14. An apparatus according to claim 13, wherein the characteristics identified by the processor relate to poles and zeros of the dynamic system.

15. An apparatus according to claim 13, wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a first state to a second state such that the dynamic system remains substantially at rest at the second state.

16. An apparatus according to claim 15, wherein the command profile includes a tail following the one or more pulses;

and wherein the dynamics cancellation constraint is based on the following equation;

17. An apparatus according to claim 13, wherein the plurality of constraints include boundary condition constraints, the boundary condition constraints including an initial time of the input and a specified change “

dy”

in the output of the system.

18. An apparatus according to claim 17, wherein the boundary condition constraints are based on the following equations:

t₀=0

19. An apparatus according to claim 13, wherein the plurality of constraints include an actuator limit constraint, the actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator.

20. An apparatus according to claim 13, wherein the processor determines the switch times based on:

(i) a command profile, u(t), comprising;

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

where “

a_j”

represents an amplitude of pulses in the function, “

t_j”

represents switch times for the pulses, “

c_j”

represents coefficients of exponential terms in a tail of the command profile, “

z_j”

represents zeros in the system, “

n”

represents a total number of switches excluding a first switch, and “

r”

represents a number of zeros in the system,

(ii) a dynamics cancellation constraint comprising;

${(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-s_{t_j}}+{\mathrm{<br><br>}}^{-s_{t_n}}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\left[c_j<br><br>t_n^{\left(q_i-1\right)}+\hspace{1em}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>\text{\hspace{1em}<br><br>}<br><br>\underset{k=1}{\overset{q_i}{\sum <br><br>}}<br><br>\left({\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_i-k\right)}]\right)<br><br>}_{s=p_i}=0,$

where “

p₁”

represents system poles, “

q₁”

represents a multiplicity of each system pole “

p₁”

, and where β

is defined as follows;

β

₁(k)=1,k=1=0,k≠

1β

_q1(k)=β

<sub>q1−

l</sub>(k)+(k−

1)β

<sub>q1−

l</sub>(k−

1),

(iii) boundary condition constraints comprising;

$\text{\hspace{1em}<br><br>}<br><br>t_0=0$$\mathrm{dy}=\underset{s-><br><br>0}{\lim }<br><br>\left(\frac{N<br><br>\left(s\right)}{D<br><br>\left(s\right)}\right)<br><br>\left({\int <br><br>}^{\left(\mathrm{rbp}\right)}<br><br>u<br><br>\left(t\right)<br><br><br><br>t\right)<br><br>\mathrm{LINESUBt}=\mathrm{\infty <br><br>},$

where N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “

rbp”

corresponds to rigid body poles in the system, and

(iv) an actuator limit constraint comprising;

u_min≦

u_tail(t)≦

u_max,

where u_min represents a minimum value of the input, u_tail(t) represents a value of a tail of the input at time “

t”

, and u_max represents a maximum value of the input.

21. An apparatus according to claim 20, wherein the switch times are determined using an optimization algorithm.

22. An apparatus according to claim 13, wherein, in a case that the identifying step identifies that the dynamic system contains poles only and no zeros, (i) the command profile comprises:

23. An apparatus according to claim 22, wherein (i) the dynamics cancellation constraint comprises:

24. An apparatus according to claim 13, wherein the command profile further comprises,

after a last one of the one or more pulses, a tail which trails off to a constant value;

and

wherein the plurality of constraints comprise insensitivity constraints that include a constraint which limits a length of the tail.

25. An apparatus according to claim 24, wherein the tail is defined by a function u_tail(t) which comprises:

26. An apparatus according to claim 25, wherein the values of the coefficients “

c_j”

are limited by;

27. An apparatus according to claim 13, further comprising the step of verifying optimality of the input which includes the switch times determined by the processor.

28. An apparatus according to claim 27, wherein the optimality of the input which includes the switch times determined in the determining step is verified based on Pontryagin'"'"'s Minimum Principle.

13. An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising:

a memory which stores computer-executable process steps; and

a processor which executes the process steps stored in the memory so as (i) to identify characteristics of the dynamic system, (ii) to select a command profile which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and (iv) to determine the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.