Fig。 2。    (a) Compass gait model and (b) Spring–mass walking model。 The dashed lines represent the vertical body excursion in these models。

model has to be elaborated for robotic implementation。 In the rest of this paper, we explore a mechanically realistic model consider- ing the theoretical spring–mass walking model and the anatomical structure of biological systems。

2。3。Model of a biped robot

Fig。 3 shows the simulation model of a biped robot investigated in this paper。 This model consists of seven body segments, two motors at the hip joints with position control, four passive joints in the knee and ankle joints, and eight linear springs。 The springs are implemented as substitutes for muscle-tendon systems, which constrain the passive joints。 A unique feature of this robot is that six of the springs are connected over two joints, referred to as biarticular muscles in biological systems (i。e。 four springs attached between the hip and the shank, two are between the thigh and the heel)。 For the sake of implementation in a real-world robotic platform, the dimension of this model is scaled down as shown in Table 1。 There are two ground contact points in a leg defined in this model。 In the simulation, we test the model in a level ground surface with a physically realistic interaction model based on a biomechanical study [15]。 The vertical ground reaction forces are approximated by nonlinear spring–damper interaction, and the horizontal forces are calculated by a sliding–stiction model。 It switches between sliding and stiction when the velocity of the foot becomes lower or higher than the specified limit。 We used 0。55 and

0。75 for the sliding and stiction friction coefficients, respectively。 For controlling the motors, we employed a simple oscillation,

because the hip joint trajectories in human walking can be roughly approximated by a sinusoid (Fig。 1)。 The angular position of the hip joint is, therefore, determined by the sinusoidal curve as follows:

Pr (t)  = A sin(2πωt) + B (1)

Pl(t) = A sin(2πωt  + π) + B (2)

where the three parameters are amplitude A, frequency ω and offset angle B。 By using this simple control scheme, we are able to evaluate how the morphological constraints can contribute to walking behavior。 This model is implemented in a planar space for the sake of simplicity, thus no rotational movement of the upper body (hip segment) is  considered。

3。Experiments

3。1。Simulation result

We constructed the proposed biped robot model in the simulation environment of Mathworks Matlab 7。01 together with the SimMechanics toolbox。

We made use of the following hip joint control parameters: A = 15 degrees, B = 5 degrees, and ω = 2 Hz。 The hip joint control parameters were determined based on the kinematic analysis  of

the human experiments (Fig。 1) for the amplitude and offset angles, and we conducted systematic search of the frequency parameter with which the model exhibits a stable periodic gait as shown in Fig。 4。 It is important to mention that this gait pattern is achieved

Fig。 3。 Biped robot model。 Only one of the two legs is shown in this figure。 The model consists of a joint controlled by a motor (represented by a black circle) and three leg segments which are connected through two passive joints (white circles)。 Two ground contact points are defined in the foot segment。

Table 1

Specification of the robot

Param。 Description Value

l1  + l2 Thigh 0。10 m

l3  + l4  + l5