Under-Actuated and Passive Mechanisms

Notes regarding Sangbae's thesis and related topics.

Theme: Design of under-actuated mechanisms for bio-inspired robotics, using iSprawl, Spinybot and Stickybot mechanisms for examples.

Formalism

We likely will not develop a formal theory of under-actuated mechanism synthesis, but we can at least come up with a relatively formal problem statement.

I propose that the basic problem is to achieve a closed trajectory in space (kind of like the leg trajectory specification problem on RiSE and the notation used to define it) plus a specification for the stiffness or compliance matrix that the mechanism will have at each point along the trajectory.

-along with the closed trajectory and stiffness matrix, it may be desirable to define a force profile as a function of the distance along the trajectory or define two trajectories, thus giving DeltaX? of the compliant mechanism and ultimately defining the force profile. -mspenko

We are only considering planar mechanisms in iSprawl, Spinybot, Stickybot, so we can limit the discussion to closed planar trajectory curves

• specify the sequence of points [x,y] that creates the curve. Alternatively, use polar coordinates r(theta) or any other two-parameter representation. The polar notation has the advantage that it automatically repeats itself every cycle. Maybe also specify an orientation at every point along the trajectory.
• specify the velocity v(s) where v = sqrt(dx/dt^2 + dy/dt^2) and s is arc length along the curve. With the velocity and the path we have a full description of the motion.
• specify the stiffness matrix associated with a point of interest on a mechanism that follows the trajectory. Formally, embed a coordinate frame P in the mechanism such that the origin of P traces the trajectory and the orientation of P keeps the local X axis tangent to the curve. When the point in which P is embedded is acted on by an external force it will exhibit a compliant behavior described by a 3x3 stiffness matrix, K(s) consisting of stiffness in the x, y and theta (rotational) directions in the plane.

The notion that a full description of behavior includes trajectory plus stiffness (or servo gains) at every point goes back to an old paper by Roger Brocket at Harvard.

• R. W.Brockett, “On the computer control of movement,” Proc. IEEE Int. Conf. Robot. Automat., 1988. -- Brocket argues that a proper specification of motion involves a sequence of tuples (u,k,t) where u is the control action (trajectory), k is the gain matrix (or stiffnesses), t is the time interval over which u and k apply.

Literature

There is a modest-sized literature on under-actuated mechanisms and compliant mechanisms in robotics. Some of the useful references include:

• S. Hirose - soft finger mechanisms that use a series of pulleys and links to provide a uniform gripping force against objects of arbitrary profile. Hirose derives the relationship for pulley torque T(s) as a function of arc length, s, along the mechanism to produce a uniform pressure. The equations and boundary conditions are nearly identical to those for the moment-curvature equations in elementary beam theory.

• Bruno Massa -- Ph.D. thesis at SSSA. Bruno was the student most responsible for the design of the under-actuated mechanisms of the "Spring Hand" at the ARTS lab. He cites various works on under-actuated mechanisms including Hirose and several publications by Gosselin and Howell. We should check these out...
  • Related material in the journal paper on SPRING hand.

Another area concerns the definition of compliance (also called admittance) or stiffness (1/compliance) matrices to match certain task requirements. The literature here derives largely from papers by Michael Peshkin and his former student Joe Schimmels. Given a task specification (e.g. for assembling a peg into a hole) can you derive the optimal stiffness matrix that a passive compliant wrist should have? And given a formulation for the ideal compliance or stiffness matrix, can you specify a set of mechanical springs that will achieve that compliance?

-- MarkCutkosky - 08 Mar 2007