Impact due to collision between the environment and the robot system occurs at the contact point during the transition from motion to constrained motion. The interacting force at the contact point is difficult to control, and the modeling and control of impact have been considered as an important issue in the field of robotics.

When human and animal contacts environment, there are many simultaneous contact points. When a bird lands on the ground, two or three toes of the bird experience simultaneous ground contact. When the human grasps an object, fingers more than two collide with an object, simultaneously.

In general contact problem like the above- mentioned cases, multiple ground contacts occur more frequently as compared to the single ground contact. Fig. 1(a) through Fig. 1(d) show that legs and feet of animals consist of large-sized bones and small-sized bones. Specially, the foot of horse only has large-sized phalange instead of small-sized toes. Fig. 1(e) through Fig. 1(f) show the skeleton of arm and leg of human. Similarly to animals, leg and arm of human also consist of largesized bone and small-sized bone in hand and foot. We will show in this paper that those articulated nature of proximal bones plays an important role to minimize or absorb impact due to collision to the environment.

Fig. 1 

Skeleton of animal and human extremities

Recently developed locomotion mechanisms with feet can be classified as a bi-pedal robot and a quadruped robot. The well-known ASIMO^[1] developed by Honda Co. Locomotion is the first successful humanoid robot. WABIAN-2 from Waseda University of Japan has a toe degree of freedom unlike ASIMO. This robot’s walking is similar to the human walking because it has one toe^[2]. H7 developed at the University of Tokyo uses various sensor information in order to get a balance of robot^[3]. Humans and animals have several toes. However, previous humanoid models have limitation in adaption to various environments because most of foot mechanisms are rectangular shaped. They can walk just on an even terrain. Though such foot mechanisms have a soft material like rubber^[3] or spring^[4] in order to absorb an impact from ground contact, they cannot structurally absorb the impact. Chung and Yi^[5] analyzed toe mechanisms of humans and animals. They claimed that multi-legged and multi-jointed foot structure were shown to be an intelligent mobility playing the role of impulse absorption and that prior to control action by actuators, passive adaptation to unknown environment is crucial. Seo and Yi^[6] analyzed the human foot structure and proposed a biomimetic foot mechanism and its closed-from dynamic model, which consists of a toe, an ankle, a heel, and springs replacing the foot muscles and tendons. It was shown through simulation that using five toes and springs, this foot could adapt to various environments in a stable manner. However, impulse model and its analysis for such multi-chained leg mechanism have not been addressed yet.

In this work, we propose a Newton-Euler based closed form internal impulse model for multi-chained robotic systems in case of multiple contacts. The structure of the multi-chained and articulated foot models is introduced in section 2. Its dynamic model and general impulse model are introduced in section 3. External impulse model is reviewed in section 4. Section 5 introduces a closed form internal impulses model for multiple contact cases. The effectiveness of the foot mechanisms is shown in section 6 through simulation and its effectiveness is verified by comparison with the simulation result of commercially available software.

Birds have many different shapes of feet as shown in Fig. 2(a) through Fig. 2(d). Song birds or perching birds (warblers, thrushes, wrens, etc.) have independent, flexible toes, with one pointing backwards, ideal for grasping perches. When perching birds sit, a tendon on the backside of the ankle automatically flexes locking their toes around the branch. With feet locked, sleeping birds do not fall. Woodpeckers have two toes pointing forwards and two backwards for climbing up, down, and sideways on tree trunks. Water birds such as duck have webbing between their toes for swimming. Gulls also have feet similar to these so they do not sink while walking in the sort sand or mud near the water’s edge. Strong-legged flightless birds, like the cassowary, protect themselves by kicking with their powerful feet and sharp claws. These feet help for running, landing, and take-off^[7].

Fig. 2 

Foot shapes of birds and mammals

Also, most mammals have various shapes to their feet as shown in Fig. 2(e) through Fig. 2(h). Canids (wolves, dog and foxes) have a complex foot structure and have four digits on each foot. The inside toe on the front foot, called the dewclaw, is much smaller and has become vestigial, which implies that it no longer has much, if any, function. On the hind foot, the first toe has been lost completely. Ungulates (deer, hippo, and pig) have even fewer toes. Artiodactyls (even-toed ungulates) have a cloven or two-part hoof in which the third and fourth digits bear the weight. In deer, the first toe has been completely lost and the second and fifth are greatly reduced. Horses are perissodactyla (odd-toed), having a non-cloven hoof. They bear the weight on the third digit ^[8]. The only evidence of an elephants five toes buried within the flesh of the foot are the toe nails. Actually, the toenails are cornfield shields in the skin and are not attached to the fingers/digits. Interestingly, not all of the toes have nails either. Overall, it is generally accepted that the African elephant has four toenails on the front feet and three on the back as opposed to the Asian elephants which have five on the front and four on the back. Its feet play a role in creating holes in which water can collect, for digging up roots from the ground, and for navigating difficult terrain^[9].

Based on the analysis of the foot structures of birds and mammals, common characteristics of such foot systems is as follows :

• (i)  foots are multi-chained system
• (ii)  each chain is articulated (many joints and links)

In the following sections, we discuss the dynamic model and impulse model of such multi-chained foot mechanisms.

In this section, we derive the dynamic model of robotic systems with a floating body. Consider a multi-chained robotic system of Fig. 3(a).

Fig. 3 

Kinematic model (a) the robot including the floating body (b) Structure of a multi-chained robot

This robot is initially floating in the air and comes to contact the ground at multiple points. The kinematic model of this robot can be depicted as Fig. 3(b), which includes virtual coordinates representing the floating behavior.

The dynamic model of the i^th open chain with single contact point is given as^[11]M1

where if iϕ_ denotes the Lagrangian coordinates of the i^th open-chain. T and F_I denote the joint torque vector and external force vector, respectively. [iIϕϕ*] and [iPϕϕϕ*] denote the inertia matrix and inertia power array of the i^th chain, respectively. G denotes the Jacobian relating the joint torque vector to the external force vector.

The dynamic model of the whole robot system with multiple contacts can be expressed as

The Lagrangian coordinates consists of the virtual joint vector (ϕ_v) and motor-controlled joint vector (ϕ_a).

When a robot system interacts with environment at multiple points, (2) can be rewritten as the following partitioned form^[10]

where T_v is a null vector because the virtual joints are not activated, and T_a denotes the joint torque vector at the motor-controlled joints. So, (3) can be decomposed into two components; one for the virtual joints and another for the active joints.

By eliminating the acceleration vector (ϕ¨_v) of the virtual joint, the dynamic model in terms of the motor-controlled joint set can be expressed as ^[4]

where

[Iaa*]=Iaa−IavIvv−1Iva,ϕ˙_aT[Paaa*]ϕ˙_a=−IavIvv−1ϕ˙_T[Pvvv*]ϕ˙_+ϕ˙_T[Paaa*]ϕ˙_,

and

∑i=1N[iGavI]T=IavIvv−1∑i=1N[viGϕvI]T−∑i=1N[aiGϕvI]T.

iF_I and N denote the impulsive external force at the i^th contact point and the number of the contact point, respectively. The relationship between the velocity of ith contact point and the motor-controlled joint velocity can be expressed asM5

where [iGavI] denotes the Jacobian, relating the i^th contact point’s velocity to the motor-controlled joint velocities.

Most generally, the impact is partially elastic in the range of 0 < e < 1 . When the coefficient of restitution ( e ) is known, the relative velocity of the colliding bodies immediately after impact can be obtained. The component of the increment of the relative velocity along a vector that is normal to the contact surface given by

where v₁ and v₂ are the absolute velocities of the colliding bodies immediately before impact, and Δv₁ and Δv₂ are the velocity increments immediately after impact.

Integration of the dynamic model given in Eq. (4) over contacting time interval givesM7

Since the position and velocities are assumed finite all the time during impact, the integral term involving ϕ˙_T[Paaa*]ϕ˙_ become zero as Δt goes to zero, as does the term involving actuation input T_a. Thus, we obtain the following simple expressionM8

where iF^_I=∫t0t0+ΔtiF_Idt is defined as the external impulse at the i^th contact point. Thus, the velocity increment of the joint variables for N contact points is

The velocity increment at the i^th contact point is obtained by the following kinematic relationship.

Assume that the robot impacts on a fixed solid surface, then the absolute velocity and the velocity increment of the fixed surface are always zero (v₂ = Δv₂ = 0). Substituting (10) into (6) and congregating them from i=1 to i=N, we haveM11

where Jij=[iGavI][Iaa*]−1[jGavI]T.

When the friction on the contacting surface is negligible, the impulse always acts at the i^th contact point along the normal vector n_i. Thus, we have

Finally, the external impulse at the i^th contact point is obtained by extracting the i^th row component of (12) as follows

where n_=[n_1Tn_2T⋯n_NT]T..

From (13), we derive the magnitude of the impulse along the normal vector at the i^th contact point as follows

which is a general external impulse model for multiple contacts. Featherstone^[10] presented an initial framework for multiple contacts distributed over multiple links based on projection matrices. The result given by (14) is based on Newton-Euler approach. Chung and Yi^[13] introduced external and internal impulse models in case of multiple collisions, which will be used in the analysis of multi-chain foot mechanisms.

When a robotic system collides with the environment, the joints of the system experience internal impulses along constrained directions of joints. The magnitude of the internal impulse cannot be measured unless we attach sensors at the contacting surface of joints. Consider a serial chain with N_br branches shown in Fig. 4. The internal impulse model for such multiple contact case is introduced in this section. Using the internal impulse model, we can analyze the internal force transmitted to links and momentum change of the system.

Fig. 4 

Link with N_br branches

The relationships between the velocity increments and the impulses in the j^th coordinate frame attached to the j^th link are obtained as

by integrating the Newton-Euler equation with respect to time. In (15) and (16), m_j and [CjI] are the mass and the moment of inertia of the j^th link, respectively. The velocity of the j^th mass center and the angular velocity of j^th link are denoted as jvCj and jωj, respectively. [j+brjR] is the rotation matrix transforming the (j+br)^th coordinate frame to the j^th coordinate frame.

The increments of the velocity of the j^th mass center and the angular velocity of the j^th link are obtained by

where [jGϕC] and [jGϕω] denote the Jacobians relating the velocity of the mass center jvCj and the angular velocity jωj of the j^th link to the joint velocity vector, respectively.

Substituting (17), (18), and (9) into the left-hand sides of (15) and (16) yieldsM20

If the k^th link experiences the external impulse at the N_k contact points, then (15) and (16) are modified asM21

(19) through (22) can be expressed as an augmented form as followsM23

where the matrix [A] given byM24

is identical to the case of a single contact. The submatrices of are defined as follows:

[AFF]=[−I321R000−I3⋱000⋱jj=1R000−I3], [ATT]=[−I321R000−I3⋱000⋱jj=1R000−I3][ATF]=[−[1r11]I3[1r12]21R000[2r22]I3⋱000⋱[j=1rj=1j]jj=1R000−[jrjj]I3].

However, the block matrices [D] and [B]with N_k contact points in the k^th link are formed as followsM25

andM26

where

vJji=mj[j0R]T[jGϕC][Iϕϕ*]−1[iGϕvI]T and wJji=[cjI][j0R]T[jGϕωI][Iϕϕ*]−1[iGϕvI]T..

Therefore, the closed form internal impulse model at the joints of the multi-chained manipulator with multiple contact points are formulated as

As noted in (27), the internal impulse at joints is directly affected by the external impulse (F^_I) and also function of robot geometry.