Simultaneous design of structure and control
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This document presents a simulation of a cart and pendulum system and describes how to simultaneously design the structure and control of the system. It includes the mass matrix, controller functions for force and torque, simulation of the system, and an optimization of the system weights and controller gains using the fmincon function in MATLAB. The goal is to minimize an objective function of the torque and counterweight mass subject to constraints on the counterweight mass.
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Consider a 4-Link robot manipulator shown below. Use the forward kine.pdf
Consider a 4-Link robot manipulator shown below. Use the forward kinematic D-H table and
write an m file that plots the manipulator. The instructions are given in the module 6. Submit
your solutions by the due date, in a single MATLAB m file.
Solution
Please give the kinetic D-H table else it would be difficult to code as we need to know the
rotation spin axis and other momentum of manipulator
Stating a general example code for manipulator with data
function X = fwd_kin(q,x)
% given a position in the configuration space, calculate the position of
% the end effector in the workspace for a two-link manipulator.
% q: vector of joint positions
% x: design vector (link lengths)
% X: end effector position in cartesian coordinates
% configuration space coordinates:
q1 = q(1); % theta 1
q2 = q(2); % theta 2
% manipulator parameters:
l1 = x(1); % link 1 length
l2 = x(2); % link 2 length
% calculate end effector position:
X = [l1*cos(q1) + l2*cos(q1+q2)
l1*sin(q1) + l2*sin(q1+q2)];
% SimulateTwolink.m uses inverse dynamics to simulate the torque
% trajectories required for a two-link planar robotic manipulator to follow
% a prescribed trajectory. It also computes total energy consumption. This
% code is provided as supplementary material for the paper:
%
% \'Engineering System Co-Design with Limited Plant Redesign\'
% Presented at the 8th AIAA Multidisciplinary Design Optimization
% Specialist Conference, April 2012.
%
% The paper is available from:
%
% http://systemdesign.illinois.edu/publications/All12a.pdf
%
% Here both the physical system design and control system design are
% considered simultaneously. Manipulator link length and trajectory
% specification can be specified, and torque trajectory and energy
% consumption are computed based on this input. It was found that maximum
% torque and total energy consumption calculated using inverse dynamics
% agreed closely with results calculated using feedback linearization, so
% to simplify optimization problem solution an inverse dynamics approach
% was used, which reduces the control design vector to just the trajectory
% design.
%
% In the conference paper several cases are considered, each with its own
% manipulator task, manipulator design, and trajectory design. The
% specifications for each of these five cases are provided here, and can be
% explored by changing the case number variable (cn).
%
% This code was incorporated into a larger optimization project. The code
% presented here includes only the analysis portion of the code, no
% optimization.
%
% A video illustrating the motion of each of these five cases is available
% on YouTube:
%
% http://www.youtube.com/watch?v=OR7Y9-n5SjA
%
% Author: James T. Allison, Assistant Professor, University of Illinois at
% Urbana-Champaign
% Date: 4/10/12
clear;clc
% simulation parameters:
p.dt = 0.0005; % simulation step size
tf = 2; p.tf = tf; % final time
p.ploton = 0; % turn off additional plotting capabilities
p.ploton2 = 0;
p.Tallow = 210; % maximum .
#Import standard math functions from math import import .docx
#Import standard math functions
from math import *
import numpy as np
import matplotlib.pyplot as plt
#Prandtl-Meyer expansion angle function:
def nu(M,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))
#Prandtl-Meyer Expansion angle for numeric extraction of Mach
number
def nu2(M,nu,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))-nu
#Derivative of Prandtl-Meyer expansion angle function
def nudif(M, y = 1.25):
return sqrt(M**2-1)/(M*(1+(y-1)*M**2/2))
#Stagnation pressure from freestream pressure.
def Pstag(Pe,M,y):
return Pe*(1+(y-1)/2*M**2)**(y/(y-1))
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagratio(M,P02,P1,y):
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2-P02/P01
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagshock(M,P1,y):#returns P02
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2*P01
#Pressure across an oblique shockwave as a function of upstream
pressure, gamma, and Mach.
def Pstatshock(P,M,y,theta):
beta = shockangle(M,y,theta)[1]
return P*(1+2*y/(y+1)*((M*sin(beta))**2-1))
#Numerical derivative for this particular function with only x
(which is M) varying.
def centraldiff(func, x, A, Ast, y, h = 0.0000001):
return (func(x+h,A,Ast,y)-func(x-h,A,Ast,y))/(2*h)
#Newton's method for four-term fuction.
def Newts(func, fin,y,A,Ast, err, iMax = 100000):
x1 = fin
it = 0
ea = np.Infinity
while ea > err and it < iMax:
x0 = x1
x1 = x0 - func(x0,A,Ast,y)/
(centraldiff(func,x0,A,Ast,y))
#x1 = x0 - func(x0,A,Ast,y)/(RPdif(x0,y))
it +=1
ea = abs((x0-x1)/x1)
return x1
#Newton's method
def Newt(Mguess,nu,gamma,err,iMax=1000):
x1 = Mguess
it = 0
ea = 100000
while ea > err and it < iMax:
x0 = x1
x1 = x0 - nu2(x0,nu,gamma)/nudif(x0,gamma)
it +=1
ea = abs((x0-x1)/x1)
return x1
#Pressure across expansion fan
def Pexpand(M1, M2, P1, y):
top = 1+(y-1)/2*M1**2
bot = 1+(y-1)/2*M2**2
return P1*(top/bot)**(y/(y-1))
#Explicit solution for beta angle. Requires theta input in
degrees.
def shockangle(M,y,thd):
th = thd*pi/180
lt1 = (M**2-1)**2
lt2 = (1+(y-1)/2*M**2)
lt3 = (1+(y+1)/2*M**2)
l = (lt1-3*lt2*lt3*tan(th)**2)**.5
xt1 = (M**2-1)**3
xt2 = lt2
xt3 = (1+(y-1)/2*M**2+(y+1)/4*M**4)
x = (xt1-9*xt2*xt3*tan(th)**2)/l**3
tanb = []
for i in range(2):
tt1 = (M**2-1)
tt2 = 2*l*cos((4*pi*float(i)+acos(x))/3)
tt3 = 3*(1+(y-1)/2*M**2)*tan(th)
tanb.append((tt1+tt2)/tt3)
return (atan(tanb[0]),atan(tanb[1]))
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Similar to Simultaneous design of structure and control
Consider a 4-Link robot manipulator shown below. Use the forward kine.pdf
Consider a 4-Link robot manipulator shown below. Use the forward kinematic D-H table and
write an m file that plots the manipulator. The instructions are given in the module 6. Submit
your solutions by the due date, in a single MATLAB m file.
Solution
Please give the kinetic D-H table else it would be difficult to code as we need to know the
rotation spin axis and other momentum of manipulator
Stating a general example code for manipulator with data
function X = fwd_kin(q,x)
% given a position in the configuration space, calculate the position of
% the end effector in the workspace for a two-link manipulator.
% q: vector of joint positions
% x: design vector (link lengths)
% X: end effector position in cartesian coordinates
% configuration space coordinates:
q1 = q(1); % theta 1
q2 = q(2); % theta 2
% manipulator parameters:
l1 = x(1); % link 1 length
l2 = x(2); % link 2 length
% calculate end effector position:
X = [l1*cos(q1) + l2*cos(q1+q2)
l1*sin(q1) + l2*sin(q1+q2)];
% SimulateTwolink.m uses inverse dynamics to simulate the torque
% trajectories required for a two-link planar robotic manipulator to follow
% a prescribed trajectory. It also computes total energy consumption. This
% code is provided as supplementary material for the paper:
%
% \'Engineering System Co-Design with Limited Plant Redesign\'
% Presented at the 8th AIAA Multidisciplinary Design Optimization
% Specialist Conference, April 2012.
%
% The paper is available from:
%
% http://systemdesign.illinois.edu/publications/All12a.pdf
%
% Here both the physical system design and control system design are
% considered simultaneously. Manipulator link length and trajectory
% specification can be specified, and torque trajectory and energy
% consumption are computed based on this input. It was found that maximum
% torque and total energy consumption calculated using inverse dynamics
% agreed closely with results calculated using feedback linearization, so
% to simplify optimization problem solution an inverse dynamics approach
% was used, which reduces the control design vector to just the trajectory
% design.
%
% In the conference paper several cases are considered, each with its own
% manipulator task, manipulator design, and trajectory design. The
% specifications for each of these five cases are provided here, and can be
% explored by changing the case number variable (cn).
%
% This code was incorporated into a larger optimization project. The code
% presented here includes only the analysis portion of the code, no
% optimization.
%
% A video illustrating the motion of each of these five cases is available
% on YouTube:
%
% http://www.youtube.com/watch?v=OR7Y9-n5SjA
%
% Author: James T. Allison, Assistant Professor, University of Illinois at
% Urbana-Champaign
% Date: 4/10/12
clear;clc
% simulation parameters:
p.dt = 0.0005; % simulation step size
tf = 2; p.tf = tf; % final time
p.ploton = 0; % turn off additional plotting capabilities
p.ploton2 = 0;
p.Tallow = 210; % maximum .
#Import standard math functions from math import import .docx
#Import standard math functions
from math import *
import numpy as np
import matplotlib.pyplot as plt
#Prandtl-Meyer expansion angle function:
def nu(M,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))
#Prandtl-Meyer Expansion angle for numeric extraction of Mach
number
def nu2(M,nu,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))-nu
#Derivative of Prandtl-Meyer expansion angle function
def nudif(M, y = 1.25):
return sqrt(M**2-1)/(M*(1+(y-1)*M**2/2))
#Stagnation pressure from freestream pressure.
def Pstag(Pe,M,y):
return Pe*(1+(y-1)/2*M**2)**(y/(y-1))
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagratio(M,P02,P1,y):
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2-P02/P01
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagshock(M,P1,y):#returns P02
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2*P01
#Pressure across an oblique shockwave as a function of upstream
pressure, gamma, and Mach.
def Pstatshock(P,M,y,theta):
beta = shockangle(M,y,theta)[1]
return P*(1+2*y/(y+1)*((M*sin(beta))**2-1))
#Numerical derivative for this particular function with only x
(which is M) varying.
def centraldiff(func, x, A, Ast, y, h = 0.0000001):
return (func(x+h,A,Ast,y)-func(x-h,A,Ast,y))/(2*h)
#Newton's method for four-term fuction.
def Newts(func, fin,y,A,Ast, err, iMax = 100000):
x1 = fin
it = 0
ea = np.Infinity
while ea > err and it < iMax:
x0 = x1
x1 = x0 - func(x0,A,Ast,y)/
(centraldiff(func,x0,A,Ast,y))
#x1 = x0 - func(x0,A,Ast,y)/(RPdif(x0,y))
it +=1
ea = abs((x0-x1)/x1)
return x1
#Newton's method
def Newt(Mguess,nu,gamma,err,iMax=1000):
x1 = Mguess
it = 0
ea = 100000
while ea > err and it < iMax:
x0 = x1
x1 = x0 - nu2(x0,nu,gamma)/nudif(x0,gamma)
it +=1
ea = abs((x0-x1)/x1)
return x1
#Pressure across expansion fan
def Pexpand(M1, M2, P1, y):
top = 1+(y-1)/2*M1**2
bot = 1+(y-1)/2*M2**2
return P1*(top/bot)**(y/(y-1))
#Explicit solution for beta angle. Requires theta input in
degrees.
def shockangle(M,y,thd):
th = thd*pi/180
lt1 = (M**2-1)**2
lt2 = (1+(y-1)/2*M**2)
lt3 = (1+(y+1)/2*M**2)
l = (lt1-3*lt2*lt3*tan(th)**2)**.5
xt1 = (M**2-1)**3
xt2 = lt2
xt3 = (1+(y-1)/2*M**2+(y+1)/4*M**4)
x = (xt1-9*xt2*xt3*tan(th)**2)/l**3
tanb = []
for i in range(2):
tt1 = (M**2-1)
tt2 = 2*l*cos((4*pi*float(i)+acos(x))/3)
tt3 = 3*(1+(y-1)/2*M**2)*tan(th)
tanb.append((tt1+tt2)/tt3)
return (atan(tanb[0]),atan(tanb[1]))
#Ma ...
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Simultaneous design of structure and control
- 1. Simultaneous design of structure and Control
- 2. Example • A cart and Pendulum
- 3. The Main Function • function [out] =pendoncartPDTracingCounterWt(mc,Kpf,Kdf,Kpt,Kpd, needplot) • % Xt, Yt is the target position of the tip of the manipulator • close all; • • M=2; • m=.78; • %mc=1.3;% counter wt • lc=.2; % counter wt position • L=1; % Link Length • l=L/2; %half length of link • J=(m*l^2)/3; • g=9.8; • d2r=pi/180;
- 4. The Mass Matrix • function MM=mass(t,x) • M1=[m+mc+M (l*m-lc*mc)*cos(x(2)); • (l*m-lc*mc)*cos(x(2)) (J+m*l^2+ Jc+mc*lc^2)]; • MM=[1 0 0 0; • 0 1 0 0; • 0 0 M1(1,1) M1(1,2); • 0 0 M1(2,1) M1(2,2)];
- 5. The Controller • function F=fx(t,x) • xd=desiredTrajectory(t,x); • F=Kpf*(xd(1)-x(1))+Kdf*(xd(3)-x(3)); • end • function T=tau(t,x) • xd=desiredTrajectory(t,x); • T=Kpt*(xd(2)-x(2))+Kdt*(xd(4)-x(4))-(m*lmc*lc)*g*sin(x(2)); • end
- 6. Simulation part • % Simulation steup Starts • x0=[-1 pi/2 0 0 ]'; % Initial condition • tf=20; % Final time • tspan=0:.005:tf; • [t,x] = ode23s(@f,tspan,x0); • % Simulation ends • N=max(size(t)); • % calculate torque • Tq=zeros(N,2); • for i=1:N • Tq(i,1)=fx(t(i),x(i,:)); • Tq(i,2)=tau(t(i),x(i,:)); • end • % Calculate final tip cocrdinate • yt_act=L*cos(x(:,2)); • xt_act=x(:,1)+L*sin(x(:,2)); • xt_des=t; • yt_des=0.25*sin(pi*t/8); • out=Tq*w1+mc*w2
- 7. Optimization • Constraint – 0<Mc<2 -Limits • Objective Function – F=w1*Tq+w2*mc • Use Optimtool for optimization – Function fmincon – Objective function pendoncartPDTracingCounterWt(mc,Kpf,Kdf,Kpt,Kpd, needplot) – Start value = [1.3,100,20,150,10,0]