On grasp based objectives in human grasping simulation

We solve optimal control problems to perform predictive human grasping simulation for a two‐finger rigid multibody model. The quality of the grasp is strongly influenced by the choice of the objective function. We investigate the quality of the grasp using objectives based on grasp matrix and the hand Jacobian. We compare the performance of optimal control grasping simulations by minimising objectives based on these grasp quality measures for a lateral grasp activity.


Grasping optimal control problem
It is very challenging to simulate the problem of human grasping due to a variety of reasons, such as the complexity of the multibody system of the hand, the choice of contact modelling and the changes in the dynamics between the prehension or reaching phase and the grasping phase. We approach these issues to reproduce human-like grasping by formulating an optimal control problem (ocp). The structure of such a problem has been detailed in [2]. We use the same approach here to perform grasping simulations with a three-dimensional, rigid, multibody model of the hand composed of the thumb and the index finger, described by redundant coordinates q. The object trajectory is described by redundant coordinates q O . The system dynamics are described through the forced discrete Euler-Lagrange (DEL) equations of motion with the discrete null-space and discrete nodal reparameterisation methods, see [1]. The DEL equations are derived via a discrete variational principle, which yields a symplectic integrator with structure preserving properties. As in [2], the contact points on the fingers are fixed to the finger phalanges while the contact points on the object surface are free to be chosen by the optimiser. The contact points on the object O get defined, when the contact is closed. The contact is modelled by a spherical joint constraint representing the hard finger model, as introduced in [4]. This constraint maintains zero relative translational velocities at the contact points between the fingers and the object. The contact forces are consistent with Coulomb's law of static friction and thus, force closure is preserved. The ocp is transcribed into a constrained optimisation problem, where a discrete objective function J d = N −1 n=0 B d is minimized subject to the DEL equations, initial and final boundary conditions and path constraints. The discrete term B d is a cost function and can represent kinematic or dynamics aspects. In this work, we describe the use of grasp quality measures in optimal control which are commonly used to define how good a particular grasp is. A detailed list for such measures can be found in [3]. Solving an ocp to obtain grasps by minimising distance between the object centroid and the contact polygon centroid has been demonstrated in [2]. Here, we use two fundamental concepts of grasping, namely the grasp matrix and the hand Jacobian to set up more objectives. To derive their relations, we firstly define the twist (vector of linear and angular velocities) for the finger joints as ν = {ν 1 , ν 2 , . . . , ν n }, where ν 1 , ν 2 , . . . , ν n are the joint velocities, as shown in Figure 1. This is related to the fingers' redundant coordinate velocities through the null-space matrix P for the open hand kinematic chain asq = P ν. We define the object twist as ν O and its relation to its redundant coordinate velocity through the object null-space

Grasp quality measures based objectives
The twists at the contact points are defined as ν C = {ν a , ν b }. The grasp matrix G T ν : ν O → ν C is a map to relate the twists at the contact points and the twist at the object centroid. It conveys the amount of input required at the contact points, to account for the changes in object twist at the centroid, while preserving a closed grasp. To ensure a proper control on this input, we extract the singular values σ Gν , and compute the ratio of the smallest to the largest singular value σ Gν ,min /σ Gν ,max , called the grasp isotropy index. A higher value for this ratio ensures that the object state can be equally well controlled in all possible directions. We use this ratio as an objective which is to be maximised. Similarly, the hand Jacobian H ν : ν → ν C is a map to relate the finger joint velocities and the twists at the contact points. Following a similar idea from the grasp matrix, we compute the singular values σ Hν and obtain the ratio σ Hν ,min /σ Hν ,max . This ratio is called the uniformity of transformation index and is to be maximised in the optimal control problem.
In relative coordinates formulations, the evaluation of the matrices G ν and H ν is quite cumbersome, as can be seen in [4]. Therein, the relation H ν ν − G T ν ν O = 0 is utilised, which in effect defines the contact closure constraint on velocity level. We evaluate these matrices in a simple way with our multibody description, by using the holonomic contact constraint g c (q, q O ) = 0. Taking a time derivative of this constraint, we get and consequently, we obtain the simple expressions for H ν and G ν .

Simulation results and conclusions
We present an example for the two-finger precision grasp, namely the lateral grasp, as simulated in [2]. This grasp is performed with one contact point on distal phalanx of the thumb and two contact points on the medial phalanx of the index finger to hold thin objects with flat surfaces, such as a credit card or a key. The grasping action is to hold a key, move it to a defined location and perform a turning motion while holding it, as shown in Figure 2.  We compare the positions of the contact points on the object O obtained by minimising object polygon centroid distance (J 1 ), from [2], with the contact points O obtained by maximising grasp isotropy index (J 2 ) and maximising uniformity of transformation (J 3 ) objectives, as shown in Figure 3. The contact points O for the J 1 objective are close to the object centroid. The contact points for the J 2 objective are away from the object centroid suggesting that the contact points near the centroid, as obtained from J 1 objective, cannot control the changes in the object trajectory as effectively as the contact points from J 2 objective. The contact points obtained with J 3 objective, however, are even further away from the centroid, when compared to both J 1 and J 2 objectives. This is consistent with the idea that holding an object away from its centroid can impart higher torques to induce the required changes in the object trajectory.
The measures derived from the grasp matrix and the hand Jacobian influence the position of the contact points on the object in expected ways. The next step to further explore the possibilities of objectives derived from grasp quality measures is to allow the optimiser to choose the contact points on the finger phalanx surfaces as well, by providing them as optimisation variables. Furthermore, an extensive analysis across multiple grasp types is to be done to investigate whether these measures behave uniformly and are independent of the grasp type.