A nonlinear Timoshenko beam formulation for modeling a tendon‐driven compliant neck mechanism

Abstract

In this paper, we consider the dynamical behavior of a tendon-driven compliant neck mechanism which is used to actuate the head of a humanoid robot as proposed by [1,2]. The neck is realized as a silicone block mounted onto the robot's torso. At the top end of the silicone block an aluminum plate interconnects the compliant neck with the head. At the same plate, tendons are attached whose actuation causes the soft and flexible block to deform thereby inducing a motion of the robot's head. For workspace design and control of the head's trajectory, a mechanical model is required which appropriately describes the entire neck-head system. We present a dynamic model of the system in which the silicone block and the head are modeled as a planar nonlinear Timoshenko beam and a rigid body, respectively. The tendon actuations are included as external configuration-dependent forces.The motion of the system is described in the three-dimensional Euclidean vector space E 3 with origin O and right-handed orthonormal coordinate frame e I i ∈ E 3 , i = {x, y, z} and takes place exclusively in the e I x -e I z -plane, see Fig. 1b. The cartesian coordinate representation of a vector a ∈ E 3 in an arbitrary orthonormal B-system rotated against the I-system is denotedThe neck made out of silicone is modeled as a planar nonlinear Timoshenko beam. According to the Timoshenko beam assumptions, the motion of the three-dimensional continuum can be described merely by the motion of a centerline and the rotations of plane rigid cross-sections attached to every point of the centerline. The centerline r = r(s, t) ∈ E 3 is a plane curve at time t parameterized by s = [0, L] ⊂ R being the arclength of the undeformed beam with length L. The cross-sections of the beam are represented by the cross-section-fixed frames e C i = e C i (s, t) ∈ E 3 , i = {x, y, z} continuously varying along the centerline and in time. The beam is fixed to the ground such that r(0, t) = 0 and e C i (0, t) = e I i , i = {x, y, z}. The inertia properties of the beam ρA and ρJ yy are the mass and moment of inertia density per unit arclength s, respectively. On top of the beam at s = L a rigid and massless plate with a width of 2b is attached as depicted in Fig. 1c. In P and R two massless tendons are connected to the plate. Both tendons are redirected by a pulley and subjected at their ends to the tensile forces λ l ≥ 0 and λ r ≥ 0, respectively. The head is modeled as a rigid body with center of mass (CoM) S, mass m, moment of inertia J and is rigidly connected to the beam in r(L, t) such that the head-fixed frame e H i (t) := e C i (L, t), i = {x, y, z}. The centerline r and the orthogonal transformation matrix A IC relating the respective coordinates according to I a = A IC C a are determined by the real-valued generalized position functions x = x(s, t), z = z(s, t) and θ = θ(s, t), i.e. I r(s, t) =   x(s, t) 0 z(s, t)   , A IC (s, t) =   cos θ(s, t) 0 sin θ(s, t) 0 1 0 − sin θ(s, t) 0 cos θ(s, t)   .(1)

Multi-input multi-output fractional-order control of an underactuated continuum mechanism

“…are computed using an iterative approach and the desired outputs y d are extracted from the solution of equation (28). A set of desired set points is computed prior to the experiments and then the steady state and transient responses of the system to this final trajectory are studied.…”

“…Rigorous mechanical models of continuum mechanisms involve nonlinear partial differential equations, 28 which result in a large number of degrees of freedom and yield that these approaches are not suitable for real-time control. 29 As outlined earlier, this work relies on a reduced nonlinear model which is based on the following assumptions.…”

Finite element formulations for constrained spatial nonlinear beam theories

“…Future research should include the application of spatial beam finite element models to real-time control applications in soft robotics [64][65][66]. Further, the constrained numerical beam models have to be validated using existing experimental data [67].…”