Gravity with Simscape Models

1. Simulink
2. Initial Simulation
3. Change the equilibrium position
4. Add external force
5. Change initial position
6. Conclusion

We would like to include gravity in a Simscape multi-body model while starting the simulation at the equilibrium.

The basic idea is to first perform a simulation of the system until it goes at it rest position and save the rest position forces and deflection in each joint.

We can then think of three solutions to start a new simulation directly at the steady state position:

Change the equilibrium position \(l_0\) of each joint such that at \(t=0\), \(k \times l_0\) is equal to the previously measured force in the joint. Here, it is suppose that we constrain the initial position of the joint to be equal to zero. Then, these forces cancel each other and the simulation starts at equilibrium.
Add an External force (or maybe the “Internal Force”) in the direction of the joint (the “resolution frame” is fixed to the solids) equal to the opposite of the previously measured force.
After the first simulation, measure the final change of position of each joint, then set the initial position to be the measured static position. Then, the simulation will start at equilibrium, but all the elements will be “deflected”.

The three solutions are tested below.

1 Simulink

open 'gravity_test.slx'

Figure 1: Simscape model used for the simulations

2 Initial Simulation

g = -10; % [m/s^2]

k = 1e3; % Stiffness [N/m]
c = 10; % Damping [N/(m/s)]

l0 = 0; % Initial wanted position [m]
leq = 0; % equilibrium position [m]

F_ext = 0; % External force [N]
F_act = 0; % Actuator force [N]

out_init = sim('gravity_test.slx')

out_init.d.Name = 'Displacement';
out_init.Fm.Name = 'Force Sensor';

figure;
subplot(1,2,1)
plot(out_init.d)
title('');
subplot(1,2,2)
plot(out_init.Fm)
title('');

<<plt-matlab>>

Figure 2: Initial Simulation (png, pdf)

3 Change the equilibrium position

l0 = 0; % Initial wanted position [m]
leq = -out_init.Fm.Data(end)/k; % equilibrium position [m]

F_ext = 0; % External force [N]
F_act = 0; % Actuator force [N]

out = sim('gravity_test.slx');

out.d.Name = 'Displacement';
out.Fm.Name = 'Force Sensor';

figure;
subplot(1,2,1)
plot(out.d)
title('');
subplot(1,2,2)
plot(out.Fm)
title('');

<<plt-matlab>>

Figure 3: Simulation after changing the equilibrium position of the joint (png, pdf)

4 Add external force

l0 = 0; % Initial wanted position [m]
leq = 0; % equilibrium position [m]

F_ext = -out_init.Fm.Data(end); % External force [N]
F_act = 0; % Actuator force [N]

out = sim('gravity_test.slx');

out.d.Name = 'Displacement';
out.Fm.Name = 'Force Sensor';

figure;
subplot(1,2,1)
plot(out.d)
title('');
subplot(1,2,2)
plot(out.Fm)
title('');

<<plt-matlab>>

Figure 4: Simulater after adding an external force applied to the solid (png, pdf)

5 Change initial position

l0 = out_init.d.Data(end); % Initial wanted position [m]
leq = 0; % equilibrium position [m]

F_ext = 0; % External force [N]
F_act = 0; % Actuator force [N]

out = sim('gravity_test.slx');

out.d.Name = 'Displacement';
out.Fm.Name = 'Force Sensor';

figure;
subplot(1,2,1)
plot(out.d)
title('');
subplot(1,2,2)
plot(out.Fm)
title('');

<<plt-matlab>>

Figure 5: Simulation after changing the initial position of the joint (png, pdf)

6 Conclusion

Three techniques:

Change the equilibrium position
- This does not change the equilibrium position of the system
Add external force
- May not be physical
Change the initial position
- The static position is not the same as the static position without gravity
- Very easy to implement

Changing the equilibrium position of each joint seem the most practical solution.