Fully implement pneumatic actuation model

examples/simulate_pneumatic_planar_pcs.py

@@ -37,7 +37,7 @@
     "r_cham_out": 2e-2 - 2e-3 * jnp.ones((num_segments,)),
     "varphi_cham": jnp.pi/2 * jnp.ones((num_segments,)),
 }
-params["D"] = 1e-3 * jnp.diag(
+params["D"] = 5e-4 * jnp.diag(
     (jnp.repeat(
         jnp.array([[1e0, 1e3, 1e3]]), num_segments, axis=0
     ) * params["l"][:, None]).flatten()
@@ -52,29 +52,88 @@
 )
 # jit the functions
 dynamical_matrices_fn = jax.jit(dynamical_matrices_fn)
-actuation_mapping_fn = auxiliary_fns["actuation_mapping_fn"]
+actuation_mapping_fn = partial(
+    auxiliary_fns["actuation_mapping_fn"],
+    forward_kinematics_fn,
+    auxiliary_fns["jacobian_fn"],
+)
 
 def sweep_actuation_mapping():
+    # evaluate the actuation matrix for a straight backbone
     q = jnp.zeros((2 * num_segments,))
     A = actuation_mapping_fn(params, B_xi, q)
-    print("A =\n", A)
+    print("Evaluating actuation matrix for straight backbone: A =\n", A)
+
+    kappa_be_pts = jnp.linspace(-jnp.pi, jnp.pi, 500)
+    sigma_ax_pts = jnp.zeros_like(kappa_be_pts)
+    q_pts = jnp.stack([kappa_be_pts, sigma_ax_pts], axis=-1)
+    A_pts = vmap(actuation_mapping_fn, in_axes=(None, None, 0))(params, B_xi, q_pts)
+    # plot the mapping on the bending strain for various bending strains
+    fig, ax = plt.subplots(num="pneumatic_planar_pcs_actuation_mapping_bending_torque_vs_bending_strain")
+    plt.title(r"Actuation mapping from $u_1$ to $\tau_\mathrm{be}$")
+    ax.plot(kappa_be_pts, A_pts[:, 0, 0], linewidth=2)
+    # shade the region where the actuation mapping is negative as we are not able to bend the robot further
+    ax.axhspan(A_pts[:, 0, 0].min(), 0.0, facecolor='red', alpha=0.5)
+    ax.set_xlabel(r"$\kappa_\mathrm{be}$ [rad/m]")
+    ax.set_ylabel(r"$\frac{\partial \tau_\mathrm{be}}{\partial u_1}$")
+    plt.grid(True)
+    plt.tight_layout()
+    plt.show()
+
+    # create grid for bending and axial strains
+    kappa_be_grid, sigma_ax_grid = jnp.meshgrid(
+        jnp.linspace(-jnp.pi, jnp.pi, 20),
+        jnp.linspace(-0.2, 0.2, 20),
+    )
+    q_pts = jnp.stack([kappa_be_grid.flatten(), sigma_ax_grid.flatten()], axis=-1)
+
+    # evaluate the actuation mapping on the grid
+    A_pts = vmap(actuation_mapping_fn, in_axes=(None, None, 0))(params, B_xi, q_pts)
+    # reshape A_pts to match the grid shape
+    A_grid = A_pts.reshape(kappa_be_grid.shape[:2] + A_pts.shape[-2:])
+
+    # plot the mapping on the bending strain
+    fig, ax = plt.subplots(num="pneumatic_planar_pcs_actuation_mapping_bending_torque_vs_axial_vs_bending_strain")
+    plt.title(r"Actuation mapping from $u_1$ to $\tau_\mathrm{be}$")
+    # contourf plot
+    c = ax.contourf(kappa_be_grid, sigma_ax_grid, A_grid[..., 0, 0], levels=100)
+    fig.colorbar(c, ax=ax, label=r"$\frac{\partial \tau_\mathrm{be}}{\partial u_1}$")
+    # contour plot
+    ax.contour(kappa_be_grid, sigma_ax_grid, A_grid[..., 0, 0], levels=20, colors="k", linewidths=0.5)
+    ax.set_xlabel(r"$\kappa_\mathrm{be}$ [rad/m]")
+    ax.set_ylabel(r"$\sigma_\mathrm{ax}$ [-]")
+    plt.tight_layout()
+    plt.show()
+
+    # plot the mapping on the axial strain
+    fig, ax = plt.subplots(num="pneumatic_planar_pcs_actuation_mapping_axial_torque_vs_axial_vs_bending_strain")
+    plt.title(r"Actuation mapping from $u_1$ to $\tau_\mathrm{ax}$")
+    # contourf plot
+    c = ax.contourf(kappa_be_grid, sigma_ax_grid, A_grid[..., 1, 0], levels=100)
+    fig.colorbar(c, ax=ax, label=r"$\frac{\partial \tau_\mathrm{ax}}{\partial u_1}$")
+    # contour plot
+    ax.contour(kappa_be_grid, sigma_ax_grid, A_grid[..., 1, 0], levels=20, colors="k", linewidths=0.5)
+    ax.set_xlabel(r"$\kappa_\mathrm{be}$ [rad/m]")
+    ax.set_ylabel(r"$\sigma_\mathrm{ax}$ [-]")
+    plt.tight_layout()
+    plt.show()
 
 
 def simulate_robot():
     # define initial configuration
-    q0 = jnp.repeat(jnp.array([5.0 * jnp.pi, 0.2])[None, :], num_segments, axis=0).flatten()
+    q0 = jnp.repeat(jnp.array([-5.0 * jnp.pi, -0.2])[None, :], num_segments, axis=0).flatten()
     # number of generalized coordinates
     n_q = q0.shape[0]
 
     # set simulation parameters
     dt = 1e-3  # time step
     sim_dt = 5e-5  # simulation time step
-    ts = jnp.arange(0.0, 2, dt)  # time steps
+    ts = jnp.arange(0.0, 7.0, dt)  # time steps
 
     x0 = jnp.concatenate([q0, jnp.zeros_like(q0)])  # initial condition
-    tau = jnp.zeros_like(q0)  # torques
+    u = jnp.array([1.2e3, 0e0])  # control inputs (pressures in the right and left chambers)
 
-    ode_fn = ode_factory(dynamical_matrices_fn, params, tau)
+    ode_fn = ode_factory(dynamical_matrices_fn, params, u)
     term = ODETerm(ode_fn)
 
     sol = diffeqsolve(

src/jsrm/systems/planar_pcs.py

@@ -256,7 +256,7 @@ def actuation_mapping_fn(
             Returns:
                 A: actuation matrix of shape (n_xi, n_xi) where n_xi is the number of strains.
             """
-            A = jnp.identity(n_xi) @ B_xi
+            A = B_xi.T @ jnp.identity(n_xi) @ B_xi
 
             return A
 
@@ -359,7 +359,7 @@ def dynamical_matrices_fn(
         # compute the stiffness matrix
         K = stiffness_fn(params, B_xi, formulate_in_strain_space=True)
         # compute the actuation matrix
-        A = actuation_mapping_fn(forward_kinematics_fn, actuation_mapping_fn, params, B_xi, q)
+        A = actuation_mapping_fn(forward_kinematics_fn, jacobian_fn, params, B_xi, q)
 
         # dissipative matrix from the parameters
         D = params.get("D", jnp.zeros((n_xi, n_xi)))
@@ -376,7 +376,7 @@ def dynamical_matrices_fn(
         D = B_xi.T @ D @ B_xi
 
         # apply the strain basis to the actuation matrix
-        alpha = B_xi.T @ A
+        alpha = A
 
         return B, C, G, K, D, alpha
 

src/jsrm/systems/pneumatic_planar_pcs.py

@@ -55,15 +55,8 @@ def actuation_mapping_fn(
             A: actuation matrix of shape (n_xi, n_act) where n_xi is the number of strains and
                 n_act is the number of actuators
         """
-        # map the configurations to strains
-        xi = B_xi @ q
-
-        # number of strains
-        n_xi = xi.shape[0]
-
         # all segment bases and tips
         sms = jnp.concat([jnp.zeros((1,)), jnp.cumsum(params["l"])], axis=0)
-        print("sms =\n", sms)
 
         # compute the poses of all segment tips
         chi_sms = vmap(forward_kinematics_fn, in_axes=(None, None, 0))(params, q, sms)
@@ -74,10 +67,20 @@ def actuation_mapping_fn(
         def compute_actuation_matrix_for_segment(
             r_cham_in: Array, r_cham_out: Array, varphi_cham: Array,
             chi_pe: Array, chi_de: Array,
-            J_pe: Array, J_de: Array, xi: Array
+            J_pe: Array, J_de: Array,
         ) -> Array:
             """
             Compute the actuation matrix for a single segment.
+            We assume that each segment contains four identical and symmetric pneumatic chambers with pressures
+            p1, p2, p3, and p4, where p1 and p3 are the right and left chamber pressures respectively, and
+            p2 and p4 are the back and front chamber pressures respectively. The front and back chambers
+            do not exert a level arm (i.e., a bending moment) on the segment.
+            We map the control inputs u1 and u2 as follows to the pressures:
+                p1 = u1 (right chamber)
+                p2 = (u1 + u2) / 2
+                p3 = u2 (left chamber)
+                p4 = (u1 + u2) / 2
+
             Args:
                 r_cham_in: inner radius of each segment chamber
                 r_cham_out: outer radius of each segment chamber
@@ -86,23 +89,45 @@ def compute_actuation_matrix_for_segment(
                 chi_de: pose of the distal end (i.e., the tip) of the segment as array of shape (3,)
                 J_pe: Jacobian of the proximal end of the segment as array of shape (3, n_q)
                 J_de: Jacobian of the distal end of the segment as array of shape (3, n_q)
-                xi: strains of the segment
             Returns:
                 A_sm: actuation matrix of shape (n_xi, 2)
             """
-            # rotation matrix from the robot base to the segment base
-            R_pe = jnp.array([[jnp.cos(chi_pe[2]), -jnp.sin(chi_pe[2])], [jnp.sin(chi_pe[2]), jnp.cos(chi_pe[2])]])
-            # rotation matrix from the robot base to the segment tip
-            R_de = jnp.array([[jnp.cos(chi_de[2]), -jnp.sin(chi_de[2])], [jnp.sin(chi_de[2]), jnp.cos(chi_de[2])]])
+            # orientation of the proximal and distal ends of the segment
+            th_pe, th_de = chi_pe[2], chi_de[2]
+
+            # compute the area of each pneumatic chamber (we assume identical chambers within a segment)
+            A_cham = 0.5 * varphi_cham * (r_cham_out ** 2 - r_cham_in ** 2)
+            # compute the center of pressure of the pneumatic chamber
+            r_cop = (
+                2 / 3 * jnp.sinc(0.5 * varphi_cham) * (r_cham_out ** 3 - r_cham_in ** 3) / (r_cham_out ** 2 - r_cham_in ** 2)
+            )
+
+            # compute the actuation matrix that collects the contributions of the pneumatic chambers in the given segment
+            # first we consider the contribution of the distal end
+            A_sm_de = J_de.T @ jnp.array([
+                [-2 * A_cham * jnp.sin(th_de), -2 * A_cham * jnp.sin(th_de)],
+                [2 * A_cham * jnp.cos(th_de), 2 * A_cham * jnp.cos(th_de)],
+                [A_cham * r_cop, -A_cham * r_cop]
+            ])
+            # then, we consider the contribution of the proximal end
+            A_sm_pe = J_pe.T @ jnp.array([
+                [2 * A_cham * jnp.sin(th_pe), 2 * A_cham * jnp.sin(th_pe)],
+                [-2 * A_cham * jnp.cos(th_pe), -2 * A_cham * jnp.cos(th_pe)],
+                [-A_cham * r_cop, A_cham * r_cop]
+            ])
+
+            # sum the contributions of the distal and proximal ends
+            A_sm = A_sm_de + A_sm_pe
 
-
-            # compute the actuation matrix for a single segment
-            A_sm = jnp.zeros((n_xi, 2))
             return A_sm
 
-        A_sms = vmap(compute_actuation_matrix_for_segment)(chi_sms, J_sms, xi)
-
-        A = jnp.zeros((n_xi, 2 * num_segments))
+        A_sms = vmap(compute_actuation_matrix_for_segment)(
+            params["r_cham_in"], params["r_cham_out"], params["varphi_cham"],
+            chi_pe=chi_sms[:-1], chi_de=chi_sms[1:],
+            J_pe=J_sms[:-1], J_de=J_sms[1:],
+        )
+        # we need to sum the contributions of the actuation of each segment
+        A = jnp.sum(A_sms, axis=0)
 
         # apply the actuation_basis
         A = A @ actuation_basis