ALS Control Loop

1. ALS Control Loop Mechanical Plant Solution Matrix Influence Matrix Fm(6) Ft(3) + Fe(3) GAIN Stiffness Steps(3) Actuator (Integrate) X X + - Fm(3) + Fm=Measured Forces Ft=Force Target on active links Fe=Force Error on active links External Forces (Gravity, Ghost)

2. Influence Matrix • The Influence Matrix can be derived from: • FEA analysis • Mathematically from simplified mirror-link-cell geometry. • Direct measurements from telescope

3. Influence Matrix – FEA Model 100N Force Input

4. Influence Matrix Calculated Link1 Link2 Link3 Link4 Link5 Link6 Infl1 1.000 -0.666 0.000 0.333 0.000 -0.666 Infl2 -0.666 1.000 -0.666 0.000 0.333 0.000 Infl3 0.000 -0.666 1.000 -0.666 0.000 0.333 Infl4 0.333 0.000 -0.666 1.000 -0.666 0.000 Infl5 0.000 0.333 0.000 -0.666 1.000 -0.666 Infl6 -0.666 0.000 0.333 0.000 -0.666 1.000 Measured Link1 Link2 Link3 Link4 Link5 Link6 Infl1 1.0000 -0.6514 0.0284 0.3117 0.0476 -0.6347 Infl2 -0.6866 1.0000 -0.6839 0.0117 0.3463 -0.0086 Infl3 0.0299 -0.6890 1.0000 -0.6728 -0.0729 0.3325 Infl4 0.3022 0.0138 -0.6252 1.0000 -0.6159 0.0101 Infl5 0.0350 0.3265 -0.0461 -0.6467 1.0000 -0.6635 Infl6 -0.7020 -0.0055 0.3448 0.0004 -0.7383 1.0000

5. Solution Matrix - Theoretical • Mathematically from the theoretical Influence Matrix, plus the loop closure constraint of: • Link4 = -Link1 • Link2 = Link3 • Link6 = Link5 Then solving for a solution matrix that will satisfy all these conditions, such as F(final)=F(initial)*M

6. Solution Matrix – Iterative Approach • A different approach to obtain the Solution Matrix was used, this method used a simplified force error vector that only solved for: • Link4 = -Link1 • Link2 = Link3 • Link6 = Link5 • Then running a loop iteratively with the actual Influence Function, until a solution was obtained, a set of initial conditions, and final values forces, were stored for each simulation ( ~20 times) • This produced a overdetermined set of equations, and the Solution Matrix was then calculated: • Solution Matrix = pinv(F(initial)) * F(final) Note Pinv = Matrix Pseudo-Inverse based on SVD Method This method produces the same solution matrix as the theoretical method when Ideal Influence Functions were used, and could be use for other cases quite easily.

7. Solution Matrix Algorithm for i=1:20, ffi(i,:)=2*(rand(1,6)-.5); % Initial Random Force Vector % Iterate to get force solution ff(1,:)=ffi(i,:); % Copy Initial Force Vector for j=2:50, f=ff(j-1,:); fe=0.5*[0 f(3)-f(2) 0 -(f(1)+f(4)) 0 f(5)-f(6)]; % Force Error - Gain=0.5 ff(j,:)=ff(j-1,:)+fe*INFL; % Iteration Loop with Influence Function end fff(i,:)=ff(j,:); % Save final forces vector end %Solve for Solution matrix (overdetermined least-squares method) M=pinv(ffi)*fff

8. Typical Iterative Solution Simulation

9. Solution Matrix Calculated From Ideal Influence Matrix Link1 Link2 Link3 Link4 Link5 Link6 Gain1 0.5000 0.2500 0.2500 -0.5000 0.2500 0.2500 Gain2 0.2500 0.5000 0.5000 -0.2500 0.0000 -0.0000 Gain3 -0.2500 0.5000 0.5000 0.2500 -0.0000 0.0000 Gain4 -0.5000 0.2500 0.2500 0.5000 0.2500 0.2500 Gain5 -0.2500 0.0000 -0.0000 0.2500 0.5000 0.5000 Gain6 0.2500 -0.0000 0.0000 -0.2500 0.5000 0.5000 Calculated From Measured Influence Matrix Link1 Link2 Link3 Link4 Link5 Link6 Gain1 0.5117 0.2358 0.2358 -0.5117 0.2279 0.2279 Gain2 0.2669 0.5002 0.5002 -0.2669 -0.0057 -0.0057 Gain3 -0.2669 0.4998 0.4998 0.2669 0.0057 0.0057 Gain4 -0.4883 0.2358 0.2358 0.4883 0.2279 0.2279 Gain5 -0.2605 0.0024 0.0024 0.2605 0.4844 0.4844 Gain6 0.2605 -0.0024 -0.0024 -0.2605 0.5156 0.5156