Diagonal control using the SVD and the Jacobian Matrix

In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model:

• Section 1.2: the model is described and its parameters are defined.
• Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
• Section 1.4: the plant is decoupled using the Jacobian matrices.
• Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
• Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
• Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
• Section 1.8: the obtained decoupled plants are compared
• Section 1.9: the diagonal controller is developed
• Section 1.10: the obtained closed-loop performances for the two methods are compared
• Section 1.11: the robustness to a change of actuator position is evaluated
• Section 1.12: the choice of the reference frame for the evaluation of the Jacobian is discussed
• Section 1.13: the decoupling performances of SVD is evaluated for a low damped and an highly damped system

The model of the gravimeter is schematically shown in Figure 1.

The parameters used for the simulation are the following:

l  = 1.0; % Length of the mass [m]
h  = 1.7; % Height of the mass [m]

la = l/2; % Position of Act. [m]
ha = h/2; % Position of Act. [m]

m = 400; % Mass [kg]
I = 115; % Inertia [kg m^2]

k = 15e3; % Actuator Stiffness [N/m]
c = 2e1; % Actuator Damping [N/(m/s)]

deq = 0.2; % Length of the actuators [m]

g = 0; % Gravity [m/s2]

%% Name of the Simulink File
mdl = 'gravimeter';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

G = linearize(mdl, io);
G.InputName  = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};

The inputs and outputs of the plant are shown in Figure 3.

More precisely there are three inputs (the three actuator forces):

And 4 outputs (the two 2-DoF accelerometers):

We can check the poles of the plant:

As expected, the plant as 6 states (2 translations + 1 rotation)

size(G)

State-space model with 4 outputs, 3 inputs, and 6 states.

The bode plot of all elements of the plant are shown in Figure 4.

Consider the control architecture shown in Figure 5.

The Jacobian matrix \(J_{\tau}\) is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass:

The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass:

We thus define a new plant as defined in Figure 5. \[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \]

\(\bm{G}_x(s)\) correspond to the \(3 \times 3\) transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5).

The Jacobian corresponding to the sensors and actuators are defined below:

Ja = [1 0 -h/2
      0 1  l/2
      1 0  h/2
      0 1  0];

Jt = [1 0 -ha
      0 1  la
      0 1 -la];

And the plant \(\bm{G}_x\) is computed:

Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName  = {'Fx', 'Fy', 'Mz'};
Gx.OutputName  = {'Dx', 'Dy', 'Rz'};

size(Gx)
State-space model with 3 outputs, 3 inputs, and 6 states.

The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6.

It is shown at the system is:

• decoupled at high frequency thanks to a diagonal mass matrix (the Jacobian being evaluated at the center of mass of the payload)
• coupled at low frequency due to the non-diagonal terms in the stiffness matrix, especially the term corresponding to a coupling between a force in the x direction to a rotation around z (due to the torque applied by the stiffness 1).

The choice of the frame in this the Jacobian is evaluated is discussed in Section 1.12.

In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required.

Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).

wc = 2*pi*10; % Decoupling frequency [rad/s]

H1 = evalfr(G, j*wc);

The real approximation is computed as follows:

D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

Now, the Singular Value Decomposition of \(H_1\) is performed: \[ H_1 = U \Sigma V^H \]

[U,S,V] = svd(H1);

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7.

The decoupled plant is then: \[ \bm{G}_{SVD}(s) = U^{-1} \bm{G}(s) V^{-H} \]

Gsvd = inv(U)*G*inv(V');

size(Gsvd)
State-space model with 4 outputs, 3 inputs, and 6 states.

The 4th output (corresponding to the null singular value) is discarded, and we only keep the \(3 \times 3\) plant:

Gsvd = Gsvd(1:3, 1:3);

The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8.

The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\):

The “Gershgorin Radii” of a matrix \(S\) is defined by: \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]

The relative gain array (RGA) is defined as:

where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix.

The obtained RGA elements are shown in Figure 10.

The RGA-number is also a measure of diagonal dominance:

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12.

Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13.

The control diagram for the centralized control is shown in Figure 14.

The controller \(K_c\) is “working” in an cartesian frame. The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.

The SVD control architecture is shown in Figure 15. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

We choose the controller to be a low pass filter: \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]

\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\)

wc = 2*pi*10;  % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; % Controller Pole [rad/s]

K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;

K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gsvd;
U_inv = inv(U);

The obtained diagonal elements of the loop gains are shown in Figure 16.

Now the system is identified again with additional inputs and outputs:

• \(x\), \(y\) and \(R_z\) ground motion
• \(x\), \(y\) and \(R_z\) acceleration of the payload.

%% Name of the Simulink File
mdl = 'gravimeter';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

G = linearize(mdl, io);
G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};

The loop is closed using the developed controllers.

G_cen = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
G_svd = lft(G, -inv(V')*K_svd*U_inv(1:3, :));

Let’s first verify the stability of the closed-loop systems:

isstable(G_cen)

ans =
  logical
   1

isstable(G_svd)

ans =
  logical
   1

The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17.

Let say we change the position of the actuators:

la = l/2*0.7; % Position of Act. [m]
ha = h/2*0.7; % Position of Act. [m]

%% Name of the Simulink File
mdl = 'gravimeter';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

G = linearize(mdl, io);
G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};

The loop is closed using the developed controllers.

G_cen_b = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
G_svd_b = lft(G, -inv(V')*K_svd*U_inv(1:3, :));

The new plant is computed, and the centralized and SVD control architectures are applied using the previously computed Jacobian matrices and \(U\) and \(V\) matrices.

The closed-loop system are still stable in both cases, and the obtained transmissibility are equivalent as shown in Figure 19.

If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobian at a point where the stiffness matrix is diagonal. A displacement (resp. rotation) of the mass at this particular point should induce a pure force (resp. torque) on the same point due to stiffnesses in the system. This can be verified by geometrical computations.

If we want to decouple the system at high frequency (determined by the mass matrix), we have tot compute the Jacobians at the Center of Mass of the suspended solid. Similarly to the stiffness analysis, when considering only the inertia effects (neglecting the stiffnesses), a force (resp. torque) applied at this point (the center of mass) should induce a pure acceleration (resp. angular acceleration).

Ideally, we would like to have a decoupled mass matrix and stiffness matrix at the same time. To do so, the actuators (springs) should be positioned such that the stiffness matrix is diagonal when evaluated at the CoM of the solid.

As the SVD is applied on a real approximation of the plant dynamics at a frequency \(\omega_0\), it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation.

Let’s do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping).

Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26.

c = 2e1; % Actuator Damping [N/(m/s)]

Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27.

c = 5e2; % Actuator Damping [N/(m/s)]

Let’s consider a parallel manipulator with several collocated actuator/sensors pairs.

System in Figure 29 will serve as an example.

We will note:

• \(b_i\): location of the joints on the top platform
• \(\hat{s}_i\): unit vector corresponding to the struts direction
• \(k_i\): stiffness of the struts
• \(\tau_i\): actuator forces
• \(O_M\): center of mass of the solid body
• \(\mathcal{L}_i\): relative displacement of the struts

The parameters are defined as follows:

l  = 1.0; % Length of the mass [m]
h  = 2*1.7; % Height of the mass [m]

la = l/2; % Position of Act. [m]
ha = h/2; % Position of Act. [m]

m = 400; % Mass [kg]
I = 115; % Inertia [kg m^2]

c1 = 2e1; % Actuator Damping [N/(m/s)]
c2 = 2e1; % Actuator Damping [N/(m/s)]
c3 = 2e1; % Actuator Damping [N/(m/s)]

k1 = 15e3; % Actuator Stiffness [N/m]
k2 = 15e3; % Actuator Stiffness [N/m]
k3 = 15e3; % Actuator Stiffness [N/m]

Let’s express \({}^Mb_i\) and \(\hat{s}_i\):

s1 = [1;0];
s2 = [0;1];
s3 = [0;1];

Mb1 = [-l/2;-ha];
Mb2 = [-la; -h/2];
Mb3 = [ la; -h/2];

Frame \(\{K\}\) is chosen such that the stiffness matrix is diagonal (explained in Section 3).

The positions \({}^Kb_i\) are then:

Kb1 = [-l/2; 0];
Kb2 = [-la; -h/2+ha];
Kb3 = [ la; -h/2+ha];

Let’s note:

• \(\bm{\mathcal{L}}\) the vector of actuator displacement:

  \begin{equation} \bm{\mathcal{L}} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix} \end{equation}

• \(\bm{\tau}\) the vector of actuator forces:

  \begin{equation} \bm{\tau} = \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} \end{equation}

• \(\bm{\mathcal{F}}_{\{O\}}\) the vector of forces/torques applied on the payload on expressed in frame \(\{O\}\):

  \begin{equation} \bm{\mathcal{F}}_{\{O\}} = \begin{bmatrix} \mathcal{F}_{\{O\},x} \\ \mathcal{F}_{\{O\},y} \\ \mathcal{M}_{\{O\},z} \end{bmatrix} \end{equation}

• \(\bm{\mathcal{X}}_{\{O\}}\) the vector of displacement of the payload with respect to frame \(\{O\}\):

  \begin{equation} \bm{\mathcal{X}}_{\{O\}} = \begin{bmatrix} \mathcal{X}_{\{O\},x} \\ \mathcal{X}_{\{O\},y} \\ \mathcal{X}_{\{O\},R_z} \end{bmatrix} \end{equation}

The Jacobian matrix can be used to:

• Convert joints velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\): \[ \dot{\bm{\mathcal{X}}}_{\{O\}} = J_{\{O\}} \dot{\bm{\mathcal{L}}} \]
• Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\): \[ \bm{\mathcal{F}}_{\{O\}} = J_{\{O\}}^T \bm{\tau} \]

with \(\{O\}\) any chosen frame.

If we consider small displacements, we have an approximate relation that links the displacements (instead of velocities):

The Jacobian can be computed as follows:

Let’s compute the Jacobian matrix in frame \(\{M\}\) and \(\{K\}\):

Jm = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
      s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
      s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];

Jk = [s1', Kb1(1)*s1(2)-Kb1(2)*s1(1);
      s2', Kb2(1)*s2(2)-Kb2(2)*s2(1);
      s3', Kb3(1)*s3(2)-Kb3(2)*s3(1)];

In the frame \(\{M\}\), the Jacobian is:

And in frame \(\{K\}\), the Jacobian is:

For a parallel manipulator, the stiffness matrix expressed in a frame \(\{O\}\) is:

where:

• \(J_{\{O\}}\) is the Jacobian matrix expressed in frame \(\{O\}\)

• \(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal

  \begin{equation} \mathcal{K} = \begin{bmatrix} k_1 & & & 0 \\ & k_2 & & \\ & & \ddots & \\ 0 & & & k_n \end{bmatrix} \end{equation}

We have the same thing for the damping matrix.

Kr = diag([k1,k2,k3]);
Cr = diag([c1,c2,c3]);

Applying the second Newton’s law on the system in Figure 29 at its center of mass \(O_M\), we obtain:

with:

• \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): \[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
• \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): \[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]
• \(\bm{\mathcal{X}}_{\{M\}}\) are displacements/rotations of the mass \(x\), \(y\), \(R_z\) expressed in the frame \(\{M\}\)
• \(\bm{\mathcal{F}}_{\{M\}}\) are forces/torques \(\mathcal{F}_x\), \(\mathcal{F}_y\), \(\mathcal{M}_z\) applied at the origin of \(\{M\}\)

Let’s use the Jacobian matrix to compute the equations in terms of actuator forces \(\bm{\tau}\) and strut displacement \(\bm{\mathcal{L}}\):

And we obtain:

The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is:

%% Mass Matrix in frame {M}
Mm = diag([m,m,I]);

Let’s note the mass matrix in the frame of the legs:

%% Mass Matrix in the frame of the struts
Ml = inv(Jm')*Mm*inv(Jm);

As we can see, the Stiffness matrix in the frame of the legs is diagonal. This means the plant dynamics will be diagonal at low frequency.

Kl = diag([k1, k2, k3]);

Cl = diag([c1, c2, c3]);

The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is defined below and its magnitude is shown in Figure 31.

Gl = inv(Ml*s^2 + Cl*s + Kl);

We can indeed see that the system is well decoupled at low frequency.

The equations of motion expressed in frame \(\{M\}\) are:

And the plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is:

with:

• \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): \[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
• \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): \[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]

%% Mass Matrix in frame {M}
Mm = diag([m,m,I]);

%% Stiffness Matrix in frame {M}
Km = Jm'*Kr*Jm;

%% Damping Matrix in frame {M}
Cm = Jm'*Cr*Jm;

The plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is defined below and its magnitude is shown in Figure 33.

%% Plant in frame {M}
Gm = inv(Mm*s^2 + Cm*s + Km);

And the system is well decoupled at high frequency (above the suspension modes).

Let’s now express the transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\). We start from:

And we make use of the Jacobian \(J_{\{K\}}\) to obtain:

And finally:

The transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is then:

The frame \(\{K\}\) has been chosen such that \(J_{\{K\}}^T \mathcal{K} J_{\{K\}}\) is diagonal.

Mk = Jk'*inv(Jm)'*Mm*inv(Jm)*Jk;

Kk = Jk'*Kr*Jk;

The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined below and its magnitude is shown in Figure 35.

Gk = inv(Mk*s^2 + Ck*s + Kk);

Jacobian matrices can be used to decouple the presented system.

Depending on the chosen frame used for the estimation of the Jacobian, different plant dynamics is obtained.

Consider a parallel manipulator with:

• \(b_i\): location of the joints on the top platform are called \(b_i\)
• \(\hat{s}_i\): unit vector corresponding to the struts
• \(k_i\): stiffness of the struts
• \(\tau_i\): actuator forces
• \(O_M\): center of mass of the solid body

Consider two frames:

• \(\{M\}\) with origin \(O_M\)
• \(\{K\}\) with origin \(O_K\)

As an example, take the system shown in Figure 36.

The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically.

The stiffness matrix in the frame \(\{K\}\) can be expressed as:

where:

• \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)

• \(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal

  \begin{equation} \mathcal{K} = \begin{bmatrix} k_1 & & & 0 \\ & k_2 & & \\ & & \ddots & \\ 0 & & & k_n \end{bmatrix} \end{equation}

The Jacobian for a planar manipulator, evaluated in a frame \(\{K\}\), can be expressed as follows:

Let’s omit the mention of frame, it is assumed that vectors are expressed in frame \(\{K\}\). It is specified otherwise.

Injecting \eqref{eq:jacobian_planar} into \eqref{eq:stiffness_formula_planar} yields:

In order to have a decoupled stiffness matrix, we have the following two conditions:

Note that we don’t have any condition on the term \(k_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x})^2\) as it is only a scalar.

Condition \eqref{eq:diag_cond_2D_1}:

• represents the coupling between translations and forces
• does only depends on the orientation of the struts and the stiffnesses and not on the choice of frame
• it is therefore a intrinsic property of the chosen geometry

Condition \eqref{eq:diag_cond_2D_2}:

• represents the coupling between forces/rotations and torques/translation
• it does depend on the positions of the joints \(b_i\) in the frame \(\{K\}\)

Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\):

And the goal is to find \({}^MO_K\) such that \eqref{eq:diag_cond_2D_2} is fulfilled:

And we have two sets of linear equations of two unknowns.

This can be easily solved by writing the equations in a matrix form:

And finally, if the matrix is invertible:

Note that a rotation of the frame \(\{K\}\) with respect to frame \(\{M\}\) would make not change on the “diagonality” of \(K_{\{K\}}\).

Consider system of Figure 37.

The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below:

ki = [5,1,2]; % Stiffnesses [N/m]
si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
bi = [[-1;0.5],[-2;-1],[0;-1]]; % Joint's positions in frame {M}

Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true:

Now, compute \({}^MO_K\):

Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);

Let’s compute the new coordinates \({}^Kb_i\) after the change of frame:

Kbi = bi - Ok;

In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\):

Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];

And the stiffness matrix:

K = Jk'*diag(ki)*Jk

Now consider the planar manipulator of Figure 38.

The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below:

ki = [1,2,1,1];
si = [[1;0],[0;1],[-1;0],[0;1]];
si = si./vecnorm(si);
h = 0.2;
L = 2;
bi = [[-L/2;h],[-L/2;-h],[L/2;h],[L/2;h]];

Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true:

ki.*si*si'

Now, compute \({}^MO_K\):

Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);

Let’s compute the new coordinates \({}^Kb_i\) after the change of frame:

Kbi = bi - Ok;

In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\):

Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];

And the stiffness matrix:

K = Jk'*diag(ki)*Jk

Let’s consider a 6dof parallel manipulator with:

• \(b_i\): location of the joints on the top platform are called \(b_i\)
• \(\hat{s}_i\): unit vector corresponding to the struts
• \(k_i\): stiffness of the struts
• \(\tau_i\): actuator forces
• \(O_M\): center of mass of the solid body

Consider two frames:

• \(\{M\}\) with origin \(O_M\)
• \(\{K\}\) with origin \(O_K\)

An example is shown in Figure 39.

The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically.

For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed in a frame \(\{K\}\) is:

where:

• \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)

• \(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal:

  \begin{equation} \mathcal{K} = \begin{bmatrix} k_1 & & & 0 \\ & k_2 & & \\ & & \ddots & \\ 0 & & & k_n \end{bmatrix} \end{equation}

The analytical expression of \(J_{\{K\}}\) is:

To simplify, we ignore the superscript \(K\) and we assume that all vectors / positions are expressed in this frame \(\{K\}\). Otherwise, it is explicitly written.

Let’s now write the analytical expressing of the stiffness matrix \(K_{\{K\}}\):

And we finally obtain:

We want the stiffness matrix to be diagonal, therefore, we have the following conditions:

Note that:

• condition \eqref{eq:diag_cond_1} corresponds to coupling between forces applied on \(O_K\) to translations of the payload. It does not depend on the choice of \(\{K\}\), it only depends on the orientation of the struts and the stiffnesses. It is therefore an intrinsic property of the manipulator.
• condition \eqref{eq:diag_cond_2} corresponds to the coupling between forces applied on \(O_K\) and rotation of the payload. Similarly, it does also correspond to the coupling between torques applied on \(O_K\) to translations of the payload.
• condition \eqref{eq:diag_cond_3} corresponds to the coupling between torques applied on \(O_K\) to rotation of the payload.
• conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} do depend on the positions \({}^Kb_i\) and therefore depend on the choice of \(\{K\}\).

Note that if we find a frame \(\{K\}\) in which the stiffness matrix \(K_{\{K\}}\) is diagonal, it will still be diagonal for any rotation of the frame \(\{K\}\). Therefore, we here suppose that the frame \(\{K\}\) is aligned with the initial frame \(\{M\}\).

Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\):

The goal is to find \({}^MO_K\) such that conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} are fulfilled.

Let’s first solve equation \eqref{eq:diag_cond_3} that corresponds to the coupling between forces and rotations:

Taking the transpose and re-arranging:

As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector, we obtain:

with:

We suppose \(k_i \hat{s}_i \hat{s}_i^T\) invertible as it is diagonal from \eqref{eq:diag_cond_1}.

And finally, we find:

If the obtained \({}^M\bm{O}_{K}\) is a skew-symmetric matrix, we can easily determine the corresponding vector \({}^MO_K\) from \eqref{eq:skew_symmetric_cross_product}.

In such case, condition \eqref{eq:diag_cond_2} is fulfilled and there is no coupling between translations and rotations in the frame \(\{K\}\).

Then, we can only verify if condition \eqref{eq:diag_cond_3} is verified or not.

Let’s define the geometry of the manipulator (\({}^Mb_i\), \({}^Ms_i\) and \(k_i\)):

ki = [2,2,1,1,3,3,1,1,1,1,2,2];
si = [[-1;0;0],[-1;0;0],[-1;0;0],[-1;0;0],[0;0;1],[0;0;1],[0;0;1],[0;0;1],[0;-1;0],[0;-1;0],[0;-1;0],[0;-1;0]];
bi = [[1;-1;1],[1;1;-1],[1;1;1],[1;-1;-1],[1;-1;-1],[-1;1;-1],[1;1;-1],[-1;-1;-1],[1;1;-1],[-1;1;1],[-1;1;-1],[1;1;1]]-[0;2;-1];

Cond 1:

ki.*si*si'

Find Ok

OkX = (ki.*cross(bi, si)*si')/(ki.*si*si');

if all(diag(OkX) == 0) && all(all((OkX + OkX') == 0))
    disp('OkX is skew symmetric')
    Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
else
    error('OkX is *not* skew symmetric')
end

% Verification of second condition
si*cross(bi-Ok, si)'

Verification of third condition

ki.*cross(bi-Ok, si)*cross(bi-Ok, si)'

Let’s compute the Jacobian:

Jk = [si', cross(bi - Ok, si)'];

And the stiffness matrix:

Jk'*diag(ki)*Jk

figure;
hold on;
set(gca,'ColorOrderIndex',1)
plot(b1(1), b1(2), 'o');
set(gca,'ColorOrderIndex',2)
plot(b2(1), b2(2), 'o');
set(gca,'ColorOrderIndex',3)
plot(b3(1), b3(2), 'o');
set(gca,'ColorOrderIndex',1)
quiver(b1(1),b1(2),0.1*s1(1),0.1*s1(2))
set(gca,'ColorOrderIndex',2)
quiver(b2(1),b2(2),0.1*s2(1),0.1*s2(2))
set(gca,'ColorOrderIndex',3)
quiver(b3(1),b3(2),0.1*s3(1),0.1*s3(2))

plot(0, 0, 'ko');
quiver([0,0],[0,0],[0.1,0],[0,0.1], 'k')

plot(Ok(1), Ok(2), 'ro');
quiver([Ok(1),Ok(1)],[Ok(2),Ok(2)],[0.1,0],[0,0.1], 'r')

hold off;
axis equal;

Equations in the \(\{M\}\) frame:

Thank to the Jacobian, we can transform the equation of motion expressed in the \(\{M\}\) frame to the frame of the legs:

And we have new stiffness and mass matrices:

with:

• The local mass matrix: \[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \]
• The local stiffness matrix: \[ K_{\{L\}} = J_{\{M\}}^{-T} K_{\{M\}} J_{\{M\}}^{-1} \]

We have that: \[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]

Therefore, we find that \(K_{\{L\}}\) is a diagonal matrix:

The dynamics from \(\tau\) to \(\mathcal{L}\) is therefore decoupled at low frequency.

The mass matrix in the frames of the legs is: \[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \] with \(M_{\{M\}}\) a diagonal matrix:

Let’s suppose \(M_{\{L\}} = \mathcal{M}\) diagonal and try to find what does this imply: \[ M_{\{M\}} = J_{\{M\}}^{T} \mathcal{M} J_{\{M\}} \] with:

We obtain:

Therefore, we have the following conditions:

The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below:

ki = [1,1,1]; % Stiffnesses [N/m]
si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
bi = [[-1; 0],[-10;-1],[0;-1]]; % Joint's positions in frame {M}

Jacobian in frame \(\{M\}\):

Jm = [si', (bi(1,:).*si(2,:) - bi(2,:).*si(1,:))'];

And the stiffness matrix in frame \(\{K\}\):

Km = Jm'*diag(ki)*Jm;

Mass matrix in the frame \(\{M\}\):

m = 10; % [kg]
I = 1; % [kg.m^2]

Mm = diag([m, m, I]);

Now compute \(K\) and \(M\) in the frame of the legs:

ML = inv(Jm)'*Mm*inv(Jm)
KL = inv(Jm)'*Km*inv(Jm)

Gm = 1/(ML*s^2 + KL);

freqs = logspace(-2, 1, 1000);
figure;
hold on;
for i = 1:length(ki)
    plot(freqs, abs(squeeze(freqresp(Gm(i,i), freqs, 'Hz'))), 'k-')
end
for i = 1:length(ki)
    for j = i+1:length(ki)
        plot(freqs, abs(squeeze(freqresp(Gm(i,j), freqs, 'Hz'))), 'r-')
    end
end
hold off;
xlabel('Frequency [Hz]');
ylabel('Magnitude');
set(gca, 'xscale', 'log');
set(gca, 'yscale', 'log');

open('drone_platform.slx');

Definition of spring parameters:

kx = 0.5*1e3/3; % [N/m]
ky = 0.5*1e3/3;
kz = 1e3/3;

cx = 0.025; % [Nm/rad]
cy = 0.025;
cz = 0.025;

We suppose the sensor is perfectly positioned.

sens_pos_error = zeros(3,1);

Gravity:

g = 0;

We load the Jacobian (previously computed from the geometry):

load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');

We initialize other parameters:

U = eye(6);
V = eye(6);
Kc = tf(zeros(6));

The plant shown in Figure 43 is identified from the Simscape model.

The inputs are:

• \(D_w\) translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
• \(\tau\) the 6 forces applied by the voice coils

The outputs are the 6 accelerations measured by the inertial unit.

%% Name of the Simulink File
mdl = 'drone_platform';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1; % Ground Motion
io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration

G = linearize(mdl, io);
G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};

% Plant
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
% Disturbance dynamics
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});

There are 24 states (6dof for the bottom platform + 6dof for the top platform).

size(G)

State-space model with 6 outputs, 12 inputs, and 24 states.

The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 44.

One can easily see that the system is strongly coupled.

Consider the control architecture shown in Figure 45. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators).

We define a new plant: \[ G_x(s) = G(s) J^{-T} \]

\(G_x(s)\) correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.

Gx = Gu*inv(J');
Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required.

Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).

wc = 2*pi*30; % Decoupling frequency [rad/s]

H1 = evalfr(Gu, j*wc);

The real approximation is computed as follows:

D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix. This can be seen on the Bode plots where the phase is close to 1. This can be verified below where only the real value of \(G_u(\omega_c)\) is shown

Now, the Singular Value Decomposition of \(H_1\) is performed: \[ H_1 = U \Sigma V^H \]

[U,~,V] = svd(H1);

The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 46.

The decoupled plant is then: \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]

Gsvd = inv(U)*Gu*inv(V');

The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\):

The “Gershgorin Radii” of a matrix \(S\) is defined by: \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]

This is computed over the following frequencies.

The relative gain array (RGA) is defined as:

where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix.

The obtained RGA elements are shown in Figure 48.

The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 49.

Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 50.

The control diagram for the centralized control is shown in Figure 51.

The controller \(K_c\) is “working” in an cartesian frame. The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.

The SVD control architecture is shown in Figure 52. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

We choose the controller to be a low pass filter: \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]

\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\)

wc = 2*pi*80;  % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; % Controller Pole [rad/s]

K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);

K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);

The obtained diagonal elements of the loop gains are shown in Figure 53.

Let’s first verify the stability of the closed-loop systems:

isstable(G_cen)

ans =
  logical
   1

isstable(G_svd)

ans =
  logical
   1

The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 54.