Consider WORMESH-II on the horizontal plane at given time instant (Fig.

10). For translational motion, three synchronous parallel pedal waves were propagated along the three parallel kinematic chains [

11], each kinematic chains has two grounding points (P

\(_{i1}\) and P

\(_{i2}\)) at each time instant, and velocities are denoted by

\(v_{i1}\) and

\(v_{i2}\) at P

\(_{i1}\) and P

\(_{i2}\) (i=1,2,3), respectively. The coordinate of COG (Center of geometry) on

\(X_{g}Y_{g}\) global frame is (X,Y). The local frame on the COG of the robot is denoted by

xy. Assumed each kinematic chains generates two wave shapes. Thus, distance between P

\(_{i1}\) and P

\(_{i2}\) is

\(\lambda\), and between P

\(_{ij}\) and P

\(_{(i+1)j}\) (i=1,2,3 and j=1.2) is

a. Assumed COG has linear velocity vector,

\(v=[v_{x}, v_{y}, 0]\) and angular velocity vector,

\(\omega _{z}=[0, 0, \omega _{z}]\). State vector

q of WORMESH-II with respect to X

\(_{g}\)Y

\(_{g}\) can be defined as

\(q=[X, Y, \theta ]^{T}\). Hence, generalised velocity vector

\(q^{.}\) can be represented by

\(\dot{q}=[\dot{X},\dot{Y},\dot{\theta }]^{T}\). Thereby,

$$\begin{aligned} \begin{bmatrix} \dot{X} \\ \dot{Y} \\ \dot{\theta } \end{bmatrix} = \begin{bmatrix} cos\theta &{} -sin\theta &{} 0 \\ sin\theta &{} cos\theta &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix} \begin{bmatrix} v_{x} \\ v_{y} \\ \omega _{z} \end{bmatrix} \end{aligned}$$

(2)

The velocity of each kinematic chain depends on its pedal wave parameters, A,

\(\omega\) and

\(\beta\) (Eq.

1). Considering overall linear disparagement along the wave propagation direction of a kinematic chain, for a unit locomotion cycle, the average velocities

\(v_{i1x}\) and

\(v_{i2x}\) in the wave propagation direction at P

\(_{i1}\) and P

\(_{i2}\) should be equal and are denoted by

\(v_{ix}\). Hence,

\(v_{i1x}\),

\(v_{i2x}\) and

\(v _{ix}\) (

\(i=1,2,3\)) can be expressed as a function of pedal wave parameters (Eq.

3).

$$\begin{aligned} v_{ix}= v_{i1x} =v_{i2x}=f(A_{i},\omega _{i}.\beta _{i}) (i=1,2,3) \end{aligned}$$

(3)

Function

f denotes the relationship between average linear velocity and travelling wave parameters. If there are no slips at ground contact points, the relationship between

\(\omega\) and

\(v_{ix}\) is proportional and linear. The relationship between

A and

\(v_{ix}\) is proportional, and

\(\beta\) controls the moving direction [

16].

$$\begin{aligned} \begin{aligned} \omega _{z}&=\ \frac{v_{11x}}{y_{COG}-a}=\frac{v_{21x}}{y_{COG}}= \frac{v_{31x}}{y_{COG}+a}= \frac{v_{x}}{y_{COG}}\\&=\ \frac{v_{y1}}{x_{COG}+b}=\frac{v_{y2}}{\lambda -b-x_{COG}}=\frac{v_{y}}{x_{COG}} \end{aligned} \end{aligned}$$

(4)

Considering rotation around an instantaneous centre of rotation

O, Eq.(

4) was derived. There is no relative motion between CM in the kinematic chain along the lateral direction to the wave propagation direction. Hence,

\(v_{i1y}=v_{y1}\) and

\(v_{i2y}=v_{y2}\) (

\(i=1,2,3\)), and all average velocity component at each grounding point can be expressed as follow for a unit locomotion cycle.

$$\begin{aligned} \begin{bmatrix} v_{1x} \\ v_{2x} \\ v_{3x} \\ v_{y1} \\ v_{y2} \end{bmatrix} = \begin{bmatrix} 1 &{} -a \\ 1 &{} 0 \\ 1 &{} a \\ 0 &{} x_{COG}+b \\ 0 &{} \lambda -b-x_{COG} \end{bmatrix} \begin{bmatrix} v_{x} \\ \omega _{z} \end{bmatrix} \end{aligned}$$

(5)

Using Eq.(

3), Eqs.(

4) and (

5), v

\(_{x}\) and

\(\omega\) can be defined as follow.

$$\begin{aligned} \begin{bmatrix} v_{x} \\ \omega _{z} \end{bmatrix} = \begin{bmatrix} \frac{f(A_{1},\omega _{1}.\beta _{1})+f(A_{2},\omega _{2}.\beta _{2})+f(A_{3}, \omega _{3}.\beta _{3})}{3}\\ \frac{f(A_{1},\omega _{1}.\beta _{1})-f(A_{3},\omega _{3}.\beta _{3})}{2a} \end{bmatrix} \end{aligned}$$

(6)

Considering Fig.

10 and Eq.(

6), locomotion kinematics of WORMESH-II can be described as follows. The overall translational displacement is resulted from the individual contribution of each active kinematic chain. Hence, the average translational velocity of COG (

\(v_{x}\)) is a function of the travelling wave parameter of individual kinematic chain (Eq.

3). By considering linear motion of individual kinematic chains,

\(v_{x}\) can be approximated to average of

\(v_{1x}\),

\(v_{2x}\), and

\(v_{3x}\). Moreover, the angular velocity,

\(\omega _{z}\), is proportional to the average velocity difference of kinematic chains, one and three. When
kinematic chain one and three have equal wave parameters, ideally robot moves in a straight line.