Start working on pneumatic planar pcs system

Author: mstoelzle

examples/simulate_pneumatic_planar_pcs.py

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+from functools import partial
+import jax
+
+jax.config.update("jax_enable_x64", True)  # double precision
+from diffrax import diffeqsolve, Euler, ODETerm, SaveAt, Tsit5
+from jax import Array, vmap
+from jax import numpy as jnp
+import matplotlib.pyplot as plt
+import numpy as onp
+from pathlib import Path
+from typing import Callable, Dict
+
+import jsrm
+from jsrm import ode_factory
+from jsrm.systems import pneumatic_planar_pcs
+
+num_segments = 1
+
+# filepath to symbolic expressions
+sym_exp_filepath = (
+    Path(jsrm.__file__).parent
+    / "symbolic_expressions"
+    / f"planar_pcs_ns-{num_segments}.dill"
+)
+
+# set parameters
+rho = 1070 * jnp.ones((num_segments,))  # Volumetric density of Dragon Skin 20 [kg/m^3]
+params = {
+    "th0": jnp.array(0.0),  # initial orientation angle [rad]
+    "l": 1e-1 * jnp.ones((num_segments,)),
+    "r": 2e-2 * jnp.ones((num_segments,)),
+    "rho": rho,
+    "g": jnp.array([0.0, 9.81]),
+    "E": 2e3 * jnp.ones((num_segments,)),  # Elastic modulus [Pa]
+    "G": 1e3 * jnp.ones((num_segments,)),  # Shear modulus [Pa]
+    "r_cham_in": 5e-3 * jnp.ones((num_segments,)),
+    "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(
+    (jnp.repeat(
+        jnp.array([[1e0, 1e3, 1e3]]), num_segments, axis=0
+    ) * params["l"][:, None]).flatten()
+)
+
+# activate all strains (i.e. bending, shear, and axial)
+# strain_selector = jnp.ones((3 * num_segments,), dtype=bool)
+strain_selector = jnp.array([True, False, True])[None, :].repeat(num_segments, axis=0).flatten()
+
+
+def simulate_robot():
+    # define initial configuration
+    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
+
+    strain_basis, forward_kinematics_fn, dynamical_matrices_fn, auxiliary_fns = (
+        pneumatic_planar_pcs.factory(sym_exp_filepath, strain_selector)
+    )
+    # jit the functions
+    dynamical_matrices_fn = jax.jit(partial(dynamical_matrices_fn))
+
+    x0 = jnp.concatenate([q0, jnp.zeros_like(q0)])  # initial condition
+    tau = jnp.zeros_like(q0)  # torques
+
+    ode_fn = ode_factory(dynamical_matrices_fn, params, tau)
+    term = ODETerm(ode_fn)
+
+    sol = diffeqsolve(
+        term,
+        solver=Tsit5(),
+        t0=ts[0],
+        t1=ts[-1],
+        dt0=sim_dt,
+        y0=x0,
+        max_steps=None,
+        saveat=SaveAt(ts=ts),
+    )
+
+    print("sol.ys =\n", sol.ys)
+    # the evolution of the generalized coordinates
+    q_ts = sol.ys[:, :n_q]
+    # the evolution of the generalized velocities
+    q_d_ts = sol.ys[:, n_q:]
+
+    # evaluate the forward kinematics along the trajectory
+    chi_ee_ts = vmap(forward_kinematics_fn, in_axes=(None, 0, None))(
+        params, q_ts, jnp.array([jnp.sum(params["l"])])
+    )
+    # plot the configuration vs time
+    plt.figure()
+    for segment_idx in range(num_segments):
+        plt.plot(
+            ts, q_ts[:, 3 * segment_idx + 0],
+            label=r"$\kappa_\mathrm{be," + str(segment_idx + 1) + "}$ [rad/m]"
+        )
+        plt.plot(
+            ts, q_ts[:, 3 * segment_idx + 1],
+            label=r"$\sigma_\mathrm{ax," + str(segment_idx + 1) + "}$ [-]"
+        )
+    plt.xlabel("Time [s]")
+    plt.ylabel("Configuration")
+    plt.legend()
+    plt.grid(True)
+    plt.tight_layout()
+    plt.show()
+    # plot end-effector position vs time
+    plt.figure()
+    plt.plot(ts, chi_ee_ts[:, 0], label="x")
+    plt.plot(ts, chi_ee_ts[:, 1], label="y")
+    plt.xlabel("Time [s]")
+    plt.ylabel("End-effector Position [m]")
+    plt.legend()
+    plt.grid(True)
+    plt.box(True)
+    plt.tight_layout()
+    plt.show()
+    # plot the end-effector position in the x-y plane as a scatter plot with the time as the color
+    plt.figure()
+    plt.scatter(chi_ee_ts[:, 0], chi_ee_ts[:, 1], c=ts, cmap="viridis")
+    plt.axis("equal")
+    plt.grid(True)
+    plt.xlabel("End-effector x [m]")
+    plt.ylabel("End-effector y [m]")
+    plt.colorbar(label="Time [s]")
+    plt.tight_layout()
+    plt.show()
+    # plt.figure()
+    # plt.plot(chi_ee_ts[:, 0], chi_ee_ts[:, 1])
+    # plt.axis("equal")
+    # plt.grid(True)
+    # plt.xlabel("End-effector x [m]")
+    # plt.ylabel("End-effector y [m]")
+    # plt.tight_layout()
+    # plt.show()
+
+    # plot the energy along the trajectory
+    kinetic_energy_fn_vmapped = vmap(
+        partial(auxiliary_fns["kinetic_energy_fn"], params)
+    )
+    potential_energy_fn_vmapped = vmap(
+        partial(auxiliary_fns["potential_energy_fn"], params)
+    )
+    U_ts = potential_energy_fn_vmapped(q_ts)
+    T_ts = kinetic_energy_fn_vmapped(q_ts, q_d_ts)
+    plt.figure()
+    plt.plot(ts, U_ts, label="Potential energy")
+    plt.plot(ts, T_ts, label="Kinetic energy")
+    plt.xlabel("Time [s]")
+    plt.ylabel("Energy [J]")
+    plt.legend()
+    plt.grid(True)
+    plt.box(True)
+    plt.tight_layout()
+    plt.show()
+
+if __name__ == "__main__":
+    simulate_robot()

src/jsrm/systems/pneumatic_planar_pcs.py

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+__all__ = ["factory", "stiffness_fn"]
+from jax import Array, vmap
+import jax.numpy as jnp
+from jsrm.math_utils import blk_diag
+import numpy as onp
+from typing import Dict, Tuple, Union
+
+from .planar_pcs import factory as planar_pcs_factory
+
+def factory(
+    *args, **kwargs
+):
+    return planar_pcs_factory(
+        *args, stiffness_fn=stiffness_fn, **kwargs
+    )
+
+def _compute_stiffness_matrix_for_segment(
+    l: Array, r: Array, r_cham_in: Array, r_cham_out: Array, varphi_cham: Array, E: Array
+):
+    # cross-sectional area [m²] of the material
+    A_mat = jnp.pi * r ** 2 + 2 * r_cham_in ** 2 * varphi_cham - 2 * r_cham_out ** 2 * varphi_cham
+    # second moment of area [m⁴] of the material
+    Ib_mat = jnp.pi * r ** 4 / 4 + r_cham_in ** 4 * varphi_cham / 2 - r_cham_out ** 4 * varphi_cham / 2
+    # poisson ratio of the material
+    nu = 0.0
+    # shear modulus
+    G = E / (2 * (1 + nu))
+
+    # bending stiffness [Nm²]
+    Sbe = Ib_mat * E * l
+    # shear stiffness [N]
+    Ssh = 4 / 3 * A_mat * G * l
+    # axial stiffness [N]
+    Sax = A_mat * E * l
+
+    S = jnp.diag(jnp.stack([Sbe, Ssh, Sax], axis=0))
+
+    return S
+
+def stiffness_fn(
+    params: Dict[str, Array],
+    B_xi: Array,
+    formulate_in_strain_space: bool = False,
+) -> Array:
+    """
+    Compute the stiffness matrix of the system.
+    Args:
+        params: Dictionary of robot parameters
+        B_xi: Strain basis matrix
+        formulate_in_strain_space: whether to formulate the elastic matrix in the strain space
+    Returns:
+        K: elastic matrix of shape (n_q, n_q) if formulate_in_strain_space is False or (n_xi, n_xi) otherwise
+    """
+    # stiffness matrix of shape (num_segments, 3, 3)
+    S = vmap(
+    _compute_stiffness_matrix_for_segment
+    )(
+        params["l"], params["r"], params["r_cham_in"], params["r_cham_out"], params["varphi_cham"], params["E"]
+    )
+    # we define the elastic matrix of shape (n_xi, n_xi) as K(xi) = K @ xi where K is equal to
+    K = blk_diag(S)
+
+    if not formulate_in_strain_space:
+        K = B_xi.T @ K @ B_xi
+
+    return K