diff --git a/Naman_LQR_Working.m b/Naman_LQR_Working.m new file mode 100644 index 0000000..f3f6e5b --- /dev/null +++ b/Naman_LQR_Working.m @@ -0,0 +1,221 @@ +% Clear workspace +clear all; close all; clc; + +% Parameters source: https://sal.aalto.fi/publications/pdf-files/eluu11_public.pdf +g = 9.81; m = 0.468; Ix = 4.856*10^-3; +Iy = 4.856*10^-3; Iz = 8.801*10^-3; + +% States: +% X1: x X4: x' +% X2: y X5: y' +% X3: z X6: z' +% X7: Pitch angle (x-axis) X10: Pitch rate (x-axis) +% X8: Roll angle (y-axis) X11: Roll rate (y-axis) +% X9: Yaw angle (z-axis) X12: Yaw rate (z-axis) + +% Inputs: Outputs: +% U1: Total Upward Force (along z-axis) Y1: Position along x axis +% U2: Pitch Torque (about x-axis) Y2: Position along y axis +% U3: Roll Torque (about y-axis) Y3: Position along z axis +% U4: Yaw Torque (about z-axis) Y4: Pitch (about x-axis) +% Y5: Roll (about y-axis) +% Y6: Yaw (about z-axis) + +% State Space Source: https://arxiv.org/ftp/arxiv/papers/1908/1908.07401.pdf +% X' = Ax + Bu +% Y = Cx + +nStates = 12; +nInputs = 4; +nOutputs = 6; + +A = [0 0 0 1 0 0 0 0 0 0 0 0; + 0 0 0 0 1 0 0 0 0 0 0 0; + 0 0 0 0 0 1 0 0 0 0 0 0; + 0 0 0 0 0 0 0 -g 0 0 0 0; + 0 0 0 0 0 0 g 0 0 0 0 0; + 0 0 0 0 0 0 0 0 0 0 0 0; + 0 0 0 0 0 0 0 0 0 1 0 0; + 0 0 0 0 0 0 0 0 0 0 1 0; + 0 0 0 0 0 0 0 0 0 0 0 1; + 0 0 0 0 0 0 0 0 0 0 0 0; + 0 0 0 0 0 0 0 0 0 0 0 0; + 0 0 0 0 0 0 0 0 0 0 0 0]; + +% Note: In paper, 1/m is in wrong spot +B = [0 0 0 0; + 0 0 0 0; + 0 0 0 0; + 0 0 0 0; + 0 0 0 0; + 1/m 0 0 0; + 0 0 0 0; + 0 0 0 0; + 0 0 0 0; + 0 1/Ix 0 0; + 0 0 1/Iy 0; + 0 0 0 1/Iz]; + +C = [1 0 0 0 0 0 0 0 0 0 0 0; + 0 1 0 0 0 0 0 0 0 0 0 0; + 0 0 1 0 0 0 0 0 0 0 0 0; + 0 0 0 0 0 0 1 0 0 0 0 0; + 0 0 0 0 0 0 0 1 0 0 0 0; + 0 0 0 0 0 0 0 0 1 0 0 0]; + +D = zeros(6,4); + +continuous = ss(A, B, C, D); +T_s = 0.05; +discrete = c2d(continuous, T_s); + +%Check if this works +impulse(discrete, 0:T_s:1); + +%We should see that U1 gets us only translation in z, U2 couples Y2 and Y4, +%U3 couples Y1 and Y5, and U4 gets us Y6 + +%% Define goals +%Goal 1: settle at 1m height <2s +x_0_up = [0, 0, -1, ... + 0, 0, 0, ... + 0, 0, 0, ... + 0, 0, 0]'; %Redefine origin! + +%Goal 2: Stabilize from a 10-degree roll and pitch with <3deg overshoot +x_0_pitch = [0, 0, 0, ... + 0, 0, 0, ... + 10, 0, 0, ... + 0, 0, 0]'; %Pitch of 10 degrees + +x_0_roll = [0, 0, 0, ... + 0, 0, 0, ... + 0, 10, 0, ... + 0, 0, 0]'; %Roll of 10 degrees + +%Goal 3: Move from position (0,0,0) to within 5 cm of (1,1,1) within 5 seconds. +x_0_trans = [-1, -1, -1, ... + 0, 0, 0, ... + 0, 0, 0, ... + 0, 0, 0]'; %Redefine origin! + +%Define Q and R for the cost function. Begin with nominal ones for all. +Q = diag([1000, 1000, 1000, ... % x, y, z + 1, 1, 100, ... % x', y', z' + 100, 100, 1, ... % roll, pitch, yaw + 1, 1, 1]); % roll', pitch', yaw' + +R = diag([10, 20, 20, 1]); % upward force, pitch torque, roll torque, yaw torque +%% Finite-Time Horizon LQR for Goal 1 + +%Calculate number of timesteps. +tSpan = 0:T_s:2; +nSteps = length(tSpan); + +%Determine gains +[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps); + +%Propagate +[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_up, K, discrete.A, discrete.B); +xlqr(3,:) = xlqr(3,:) + 1; +%Plot +plot_states(xlqr, tSpan); + +%% Finite-Time Horizon LQR for Goal 2 + +%Calculate number of timesteps. +tSpan = 0:T_s:2; +nSteps = length(tSpan); + +%Determine gains +[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps); + +%Pitch Goal +%Propagate +[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_pitch, K, discrete.A, discrete.B); + +%Plot +plot_states(xlqr, tSpan); + +%Roll Goal +%Propagate +[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_roll, K, discrete.A, discrete.B); + +%Plot +plot_states(xlqr, tSpan); + +%% Finite-Time Horizon For Goal 3 + +%Calculate number of timesteps. +tSpan = 0:T_s:5; +nSteps = length(tSpan); + +%Determine gains +[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps); + +%Pitch Goal +%Propagate +[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_trans, K, discrete.A, discrete.B); +xlqr(1:3,:) = xlqr(1:3,:) + 1; + +%Plot +plot_states(xlqr, tSpan); + +%% Helper Functions + +function [K, P] = LQR_LTI(A, B, Q, R, nSteps) + %Set P up + P = zeros(size(Q, 1), size(Q, 2), nSteps); + %Initial value of P + P(:, :, nSteps) = 1/2 * Q; + %Set K up, initial K is 0, so this is fine. + K = zeros(length(R), length(Q), nSteps); + + for i = nSteps-1:-1:1 + P_ = P(:,:, i+1); + + K(:, :, i) = ( 1/2 * R + B' * P_ * B )^(-1) * B' * P_ * A; + P(:, :, i) = A' * P_ * ( A - B * K(:, :, i) ) + Q * 1/2; + end +end + +function [ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0, K, A, B) + %Set up for propagation + ulqr = zeros(nInputs, nSteps); + xlqr = zeros(nStates, nSteps); + xlqr(:, 1) = x_0; + + for i = 1:(nSteps - 1) + ulqr(:,i) = K(:,:,i) * xlqr(:,i); + xlqr(:,i+1) = (A*xlqr(:, i) - B*ulqr(:, i)); + end +end + +function plot_states(xlqr, tSpan) + figure(); + subplot(1, 2, 1); + plot(tSpan, xlqr(1, :), '-r', 'LineWidth', 2); + hold on; + plot(tSpan, xlqr(2, :), '-g'); + plot(tSpan, xlqr(3, :), '-b'); + plot(tSpan, xlqr(4, :), '--r'); + plot(tSpan, xlqr(5, :), '--g'); + plot(tSpan, xlqr(6, :), '--b'); + legend('x', 'y', 'z', 'x`', 'y`', 'z`'); + title("Translations(-) and Velocities (--)"); + xlabel("Time(s)"); + ylabel("Displacement (m)"); + + subplot(1, 2, 2); + plot(tSpan, xlqr(7, :), '-r'); + hold on; + plot(tSpan, xlqr(8, :), '-g'); + plot(tSpan, xlqr(9, :), '-b'); + plot(tSpan, xlqr(10, :), '--r'); + plot(tSpan, xlqr(11, :), '--g'); + plot(tSpan, xlqr(12, :), '--b'); + legend('Pitch (about x)', 'Roll (about y)', 'Yaw (about z)', 'Pitch Rate', 'Roll Rate', 'Yaw Rate'); + title("Angular Displacements(-) and Velocities(--)"); + xlabel("Time(s)"); + ylabel("Displacement (deg)"); +end \ No newline at end of file