Rolling Biped Polynomial Motion Planning

Abstract

This work discloses a kinematic control model to describe the geometry of motion of a two-wheeled biped’s limbs. Limb structure is based on a four-bar linkage useful to alleviate damping motion during self-balance. The robot self-balancing kinematics geometry combines with user-customized polynomial vector fields. The vector fields generate safe reference trajectories. Further, the robot is forced to track the reference path by a model-based time-variant recursive controller. The proposed formulation showed effectiveness and reliable performance through numerical simulations.

1. Introduction

Motion planning is an essential function and a critical aspect in robotics engineering. Motion planning allows increasing the robot’s degree of autonomy. Fundamentally a wide area of robotic tasks needs planning models such as: transportation and vehicular technology, service robotics, search, exploration, surveillance, biomedical robotic applications, spatial deployment, industry, and so forth. Moreover, motion planning is inherently impacted by the degree of holonomy and kinematic constraints in all robotic modalities: robot arms, legged robots, rolling platforms, marine/underwater vehicles, aerial robots, and including their end effectors. Depending on the robotic application, motion planning is designed either global or local. When the robot has an environmental map in advance, it is called global planning even with possibility to globally optimize routes. Alternatively, when there is only robot’s local sensor data and the whole environment is unknown, it uses feedback from local observations. Planning methods can be generalized into four types: deterministic (based on mathematical numeric/analytic functions and models) [1, 2, 3], stochastic (recursive numerical methods based on probabilistic uncertainties) [4, 5, 6], heuristic (algorithms based on logical control and human-heuristic decision-making) [7, 8, 9, 10], and mixed planning methods [11, 12, 13].

In this chapter, a kinematic motion/path planning method for path tracking of an inverted pendulum self-balancing rolling biped is deduced and discussed. This work is focused on the rolling biped motion modeling and simulation of the robot shown in Figure 1. The principal component of a rolling biped is self-balancing by controlling its pitch motion through in-wheel motors that allow rolling motion (inverted-pendulum-like). The robot’s yaw motion is accomplished by the angular velocity resulting from the differential lateral speeds, which is a nonholonomic constrained model. The robot’s design is purposed for missions to collect solid garbage in outdoors (a park), similar to other works [14]. Nevertheless, the robotic mission/task is out of this chapter’s scope, instead a detailed geometry of motion is described in three parts: (i) motion planning for biped’s balance, (ii) navigational path generation, and (iii) path tracking control. The work [15] proposed a balancing and dynamic speed control of a unicycle robot based on variable structure and linear quadratic regulator to follow a desired trajectory. The work [16] modeled a wheeled bipedal robot with analytic solutions of closed-form expressions in kinematic control loops. The work [17] reported a self-balancing two-wheeled robot with a manipulator on-board, using auto-balancing system to maintain force equilibrium. The work [18] applied a proportional integral derivative (PID) control and active disturbance rejection control to balance and steer a two-wheeled self-balancing robot modeled by Lagrange formula. In [19], an adaptive robust control of a self-balancing two-wheeled underactuated robot to estimate uncertainty bound information, using deterministic system performance by Lyapunov method, was reported. In [20], a navigational two-wheeled self-balancing robot control using a PD-PI controller based on the Kalman filter algorithm was reported. Similarly, [21] used variable structure combining proportional integral differential controllers for balance and locomotion deriving a kinematic model based on the center of gravity constraint. Lagrangian-based with Kane’s approach for dynamic balancing was reported in [22]. Moreover, [23] conducted a study using model predictive control for trajectory tracking of an inverted-pendulum wheeled robot. The work [24] reported a self-balancing robot controller using Euler-Lagrange and geometric control, and planar motion is controlled by logarithmic feedback and Lie group exponential coordinates. The work [25] developed a balancing and trajectory tracking system for an inverted-pendulum wheeled robot using a Lagrange-based backstepping structure variable method. This work’s main contributions are an original design of limbs based on four-bar linkages to alleviate damping motion yielded from irregular terrains, from which a kinematic balancing condition is deduced. Further, polynomial vector fields with limit conditions are deduced from user-customized interpolation functions as path-generator models to yield safe routes in advance. Moreover, a recursive time-varying kinematic controller forces the robot to track resulting routes. The proposed system is demonstrated at the level of simulation. This chapter is organized in the following sections. Section 2 deduces the limb kinematics and its effects in the biped’s balance. Section 3 presents the polynomial approach to trajectory generation by directional fields. Section 4 describes a navigation recursive controller for path tracking control. Finally, Section 5 presents the work’s conclusions.

Figure 1.

Rolling biped platform views. (a) Isometric view. (b) Front-view. (c) Onboard robotic arm.

2. Balancing motion planning model

The main issue of an inverted-pendulum-like rolling biped is its capability to self-balance by controlling its pitch angle through the wheels velocity longitudinally (Figure 2(b)). This section deduces the balancing kinematics of the limb’s planar linkage shown in Figure 2(a). As a difference from other biomechanical inspired muscle-tendon limbs [26], in this work each limb is comprised of a four-bar linkage operating as a double crank, where bars r and d are linked by a coupling link l with limited rotary angles.

Figure 2.

Rolling biped kinematic constrains. (a) Limb’s planar linkage parameters. (b) Balancing parameters.

The expressions provided in Proposition 2.1 are obtained from deductions in Appendix A. The passive joint point ${\mathbf{P}}_1={\left(x_1,y_1\right)}^{\top }$ mutually depends on the analytic model of the joint located at ${\mathbf{P}}_2={\left(x_2,y_2\right)}^{\top }$ to describe a rotary planar motion, proposed by

Proposition 2.1 Limbs motion estimation. The four-bar linkage’s passive jointP₂is analytically described by.

$${\mathbf{P}}_2=d\left(\begin{array}{l}cos\left(\theta \right)\\ sin\left(\theta \right)\end{array}\right)+\left(\begin{array}{l}\mathrm{\Delta }x\\ \mathrm{\Delta }y\end{array}\right).$$E1

Therefore, based on Definitions 4.1–4.3 of Appendix A, the model solutionP₂is

$$\left(\begin{array}{l}{x}_2\\ y_2\end{array}\right)=d\left(\begin{array}{l}cos\left(2arctan\left(\frac{-Q\pm \sqrt{Q^2-4\mathit{PR}}}{2P}\right)\right)\\ sin\left(2arctan\left(\frac{-Q\pm \sqrt{Q^2-4\mathit{PR}}}{2P}\right)\right)\end{array}\right)+\left(\begin{array}{l}\mathrm{\Delta }x\\ \mathrm{\Delta }y\end{array}\right)$$E2

From Eq. (56) deduced in Appendix A, there are two possible solutions for θ, described as opened or crossed motion. For this work, the type of motion should be $\sqrt{Q^2-4\mathit{PR}}\forall Q^2\le 4\mathit{PR}$ that must be satisfied. Therefore, by using expressions (47), (52)–(56), the plots shown in 3 are obtained.

Furthermore, the wheels position is tracked by P_f (Figure 2(a)). This design assumes the robot’s center of gravity (cog) located at the same length $f\equiv a$ (both in Figure 3) that is projected within the biped’s body. Therefore, the cog is described by Definition 2.1:

Figure 3.

Limb motion planning produced with parameters r = 0.15 m, d = 0.15 m, l = 0.03 m, Δx = −0.07 m, Δy = 0.05387 m, $\forall 201°\le \varphi \le 236°$.

Definition 2.1 Robot’s center of gravity (cog). The robot’s cog is assumed to be located at Cartesian body’s position,

$$x_{\mathit{cog}}=\left(\frac{x_2-x_1}{2}+a\right)cos\left(\pi +arctan\left(\frac{y_2-y_1}{x_2-x_1}\right)\right)$$E3

and

$$y_{\mathit{cog}}=\left(\frac{y_2-y_1}{2}+a\right)sin\left(\pi +arctan\left(\frac{y_2-y_1}{x_2-x_1}\right)\right).$$E4

Hence, in accordance to Definition 2.1, it follows that the robot’s falling speed v_f due to vertical unbalance is

$$v_f=\sqrt[2]{{\left(\frac{{\mathit{dx}}_{\mathit{cog}}}{\mathit{dt}}\right)}^2+{\left(\frac{{\mathit{dy}}_{\mathit{cog}}}{\mathit{dt}}\right)}^2}.$$E5

Taking into account that the robot is a dual differential drive kinematic structure, where v_R and v_L are the right-sided and left-sided velocities, respectively

$$\frac{\mathit{ds}}{\mathit{dt}}=\frac{r}{2}\left(\frac{d{\phi }_R}{\mathit{dt}}+\frac{d{\phi }_L}{\mathit{dt}}\right),$$E6

where s is the robot displacement over the ground, and r is the wheel’s radius [m]. Likewise, the dual differential velocity is

$$v_{\mathit{\text{diff}}}=\frac{d{\phi }_R}{\mathit{dt}}-\frac{d{\phi }_L}{\mathit{dt}}.$$E7

Hence, the robot’s angular velocity $\omega \left[\mathrm{rad}/\mathrm{s}\right]$ in terms of its differential speed v_diff and constrained by its wheels lateral baseline distance b_l and v_diff is substituted to yield:

$$\omega =\frac{2v_{\mathit{\text{diff}}}}{b_l}=\frac{2r_w}{b_l}\left(\frac{d{\phi }_R}{\mathit{dt}}-\frac{d{\phi }_L}{\mathit{dt}}\right).$$E8

The biped’s falling angle and angular falling speed are $\lambda \left[\mathrm{rad}\right]$ and $\dot{\lambda }\left[\mathrm{rad}/\mathrm{s}\right]$, respectively. Let v_f [m/s] be the falling velocity affecting the robot’s balance

$$v_f={\mathrm{\ell }}_b{\dot{\lambda }}_t,$$E9

which makes the robot’s pitch turn around the wheel’s center. Unbalancing velocity v_u [m/s] is an horizontal component such that

$$v_u=v_{f}cos\left(\frac{\pi }{2}-\lambda \right)={\mathrm{\ell }}_b\dot{\lambda }cos\left(\frac{\pi }{2}-\lambda \right).$$E10

The wheel’s axis point moves at balancing speed v_b that is parallel to the ground. Wheel’s tangential speed is $v_w=r\dot{\phi }$, and linear motion speed is the balancing velocity underneath the biped’s body at the wheel’s axis,

$$v_b=\frac{v_w}{2}=\frac{r{\dot{\phi }}_w}{2}$$E11

Therefore, the balancing condition is described by Definition 2.2:

Definition 2.2 Kinematic balancing condition. The robot’s vertical equilibrium condition is assumed for balancing and unbalancing velocities of equal magnitudes (or very approximated). Thus,

$$v_u-v_b=0,$$E12

and redefining, let${\dot{\phi }}_B$be the wheel’s balancing angular speed,

$${\mathrm{\ell }}_b\dot{\lambda }cos\left(\frac{\pi }{2}-\lambda \right)-\frac{r_w{\dot{\phi }}_B}{2}=0.$$E13

Therefore, it is of interest to find the wheel’s rotary speed that balances the biped motion and from Definition 2.2, the following Proposition 2.2 arises,

Proposition 2.2 Robot’s balancing speed. The wheel’s rolling velocity to satisfy the robot’s body balance is proposed as

$${\dot{\phi }}_B=\frac{2{\mathrm{\ell }}_b}{r_w}{\dot{\lambda }}_{t}cos\left(\frac{\pi }{2}-\lambda \right).$$E14

and if and only if${\dot{\phi }}_R={\dot{\phi }}_L$must exist, assuming that in such period of time, the robot’s yaw${\omega }_t\approxeq 0$allowing longitudinal balancing equilibrium. Thus,

$$v_t=\left\{\begin{array}{l}r{\dot{\phi }}_B,{\dot{\phi }}_R\approx {\dot{\phi }}_L,v_u-v_b\ne 0\\ \frac{r}{2}\left({\dot{\phi }}_R+{\dot{\phi }}_L\right),v_u-v_b\approx 0\end{array}\right.$$E15

Thus, let us redefine the state variables as $x_1=\lambda$, $x_2={\dot{x}}_1$, ${\dot{x}}_2=x_2^{2}tan\left(\frac{\pi }{2}-x_1\right)$. For an stability analysis for the balancing case when $v_t=r_w{\dot{\phi }}_B$, and equilibrium condition ${\ddot{\phi }}_B=0$ occurs,

$$\frac{2{\mathrm{\ell }}_b}{2}\left({\dot{x}}_{2}cos\left(\frac{\pi }{2}-x_1\right)+{{\dot{x}}_2}^{2}sin\left(\frac{\pi }{2}-x_1\right)\right)=0$$E16

and dropping off the highest-order derivative of the system

$$\ddot{\lambda }=x_2^{2}tan\left(\frac{\pi }{2}-x_1\right)$$E17

The system total energy $E_k+E_p$ (kinetic plus potential) is a positive function, which is used as a Lyapunov candidate function, where the robot’s translation motion $r_w{\dot{\phi }}_B$ for equilibrium is

$$E_T=E_k+E_p=\frac{1}{2}{\mathit{mr}}_w^2{\dot{\phi }}_B^2+\mathit{mg}\frac{{\mathrm{\ell }}_b}{2}\left(1-cos\left(\lambda \right)\right)$$E18

and substituting ${\dot{\phi }}_B$,

$$E_T=2m{\mathrm{\ell }}_b^2{\dot{\lambda }}^{2}cos\left(\frac{\pi }{2}-\lambda \right)+\mathit{mg}\frac{{\mathrm{\ell }}_b}{2}\left(1-cos\left(\lambda \right)\right).$$E19

For $V\left(x\right)=E_T$ and finding out that v(x) and $\dot{V}\left(x\right)$ fulfill $\dot{V}\left(0\right)=0$. In addition, assuming that $-2\pi \ll \lambda \ll 2\pi$, then $\dot{V}\left(0\right)>\in D-0$. Assuming that $V:D\to \mathbb{R}$ is a continuous differentiable function. Finally, the following Lyapunov criterion is satisfied (20),

$$\left(\frac{\partial V\left(x\right)}{\partial x_1},\frac{\partial V\left(x\right)}{\partial x_2}\right)\cdot {\left(x_2,{\dot{x}}_2\right)}^{\top }=0$$E20

3. Polynomial vector fields

This section introduces the proposed path planning strategy to dynamically generate local paths. Two polynomial models are enhanced as vector fields to exert attractive and repulsive robot’s accelerations. In principle, the desired acceleration functions are designed from the interpolating attractive/repulsive accelerations with respect to (w.r.t.) distance. The method used to fit points is the Lagrange formula:

$$a\left(\delta \right)=\sum _i\left(\prod _j\frac{\delta -{\delta }_i}{{\delta }_i-{\delta }_j}\right)a_i,$$E21

where the human-user establishes a desired numerical acceleration a_i w.r.t. a desired distance δ_i, from either goal or obstacle. Therefore, the following general cubic polynomial forms attractive a_A and repulsive a_R are obtained with their respective numeric coefficients λ_i and β_i,

$$a_A\left(\delta \right)={\lambda }_0+{\lambda }_1\delta +{\lambda }_2{\delta }^2+{\lambda }_3{\delta }^3,a_R\left(\delta \right)={\beta }_0+{\beta }_1\delta +{\beta }_2{\delta }^2+{\beta }_3{\delta }^3$$E22

It follows that, Definition 3.1 establishes the acceleration path planning model toward a goal of interest.

Definition 3.1 Planning toward a goal. Given the kinematic parameters${\lambda }_0=0$,${\lambda }_1=\frac{18}{75}\left[s^{-2}\right]$,${\lambda }_2=\frac{18}{75}\left[m^{-1}{s}^{-2}\right]$,${\lambda }_3=\frac{3}{75}\left[m^{-2}{s}^{-2}\right]$, the attraction path model$-{\nabla }_{\delta }\kappa f^A\left(\delta \right)$is defined by

$$f^A\left({\delta }_g\right)=-{\nabla }_{\delta }\frac{3}{75}\left(6{\delta }_g+6{\delta }_g^2-{\delta }_g^3\right).$$E23

Assuming that the distance between the goal and the robot${\delta }_g=\sqrt[2]{x^2+y^2}$, such that$x\doteq x_g-x_r$and$y\doteq y_g-y_r$.

Similarly, Definition 3.2 establishes the acceleration path planning model that avoids obstacles zones.

Definition 3.2 Planning obstacles avoidance. Given the kinematic parameters${\beta }_0=5$,${\beta }_1=\frac{109}{440}$,${\beta }_2=-\frac{18}{55}\left[m^{-1}{s}^{-2}\right]$,${\beta }_3=\frac{3}{110}\left[m^{-2}{s}^{-2}\right]$, the avoidance path model$-{\nabla }_{\delta }{f}^R\left(\delta \right)$is defined by

$$f^R\left({\delta }_o\right)=-{\nabla }_{\delta }{\kappa }_R\left(5+\frac{109}{440}{\delta }_o-\frac{18}{55}{\delta }_o^2+\frac{3}{11}{\delta }_o^3\right),$$E24

Assuming that the distance between the obstacle and the robot${\delta }_0=\sqrt[2]{x^2+y^2}$, such that$x\doteq x_o-x_r$and$y\doteq y_o-y_r$.

Figure 4(a) shows the robot’s instantaneous acceleration toward a goal of interest. The attraction acceleration starts when the goal-robot distance ${\delta }_g<9\mathrm{m}$. The planner allows start from $f^A\approxeq 0$ to realistically provide speeds physically possible. Figure 4(b) shows the robot’s instantaneous acceleration away from obstacles. The avoidance acceleration starts when the obstacle-robot distance ${\delta }_o<8\mathrm{m}$. This avoidance planner $f^R$ is faster than $f^A$ to increase confidence against obstacles.

Figure 4.

Navigational robot’s motion planning, desired acceleration w.r.t. distance. (a) Goal attractive motion. (b) Obstacle avoidance motion.

Therefore, extending to two-dimension Cartesian space, let us deduce the following algebraic process for the goal-attraction planner $f^A$:

$$\frac{\partial f^A}{\partial x}=-{\kappa }_A\left(\frac{6x}{\sqrt[2]{x^2+y^2}}+12x-\frac{3x}{2}\sqrt[2]{x^2+y^2}\right),$$E25

as well as

$$\frac{\partial f^A}{\partial y}=-{\kappa }_A\left(\frac{6y}{\sqrt[2]{x^2+y^2}}+12y-\frac{3y}{2}\sqrt[2]{x^2+y^2}\right).$$E26

Similarly, for the robot’s acceleration to avoid obstacles, let us develop the following $f^R$, by substituting the functional form of ${\delta }_o$ into the gradient function

$$f^R\left(x,y\right)=-\nabla {\kappa }_R\left(5+\frac{109}{440}\sqrt[2]{x^2+y^2}-\frac{18}{55}\left(x^2+y^2\right)+\frac{3}{110}{\left(x^2+y^2\right)}^{3/2}\right),$$E27

subsequently by applying the gradient operator,

$$\frac{\partial f^R}{\partial x}=-\frac{109x}{440\sqrt[2]{x^2+y^2}}+\frac{18x}{55}-\frac{9x}{110}\sqrt[2]{x^2+y^2}$$E28

and

$$\frac{\partial f^R}{\partial y}=-\frac{109y}{440\sqrt[2]{x^2+y^2}}+\frac{18y}{55}-\frac{9y}{110}\sqrt[2]{x^2+y^2}$$E30

Thus, by substituting terms in both local planners, the general attractive motion planning is a negative function with amplitude coefficient ${\kappa }_A=\frac{3}{75}\left[s^{-2}\right]$ is represented by

$$\left(\begin{array}{l}\frac{\partial f^A}{\partial x}\\ \frac{\partial f^A}{\partial y}\end{array}\right)=-{\kappa }_A\left(\frac{6}{{\delta }_t}+12-\frac{3}{2}{\delta }_g\right)\left(\begin{array}{l}{x}_g-x_t\\ y_g-y_t\end{array}\right)$$E31

The attractive field is a positive function working within limits $0\le {\delta }_g<9$, thus

$$\underset{{\delta }_g\to 9}{lim}{f}^R\left({\delta }_g\right)=0;\text{and}\underset{{\delta }_g\to 6}{lim}{f}^R\left({\delta }_g\right)=3.75$$E32

and the repulsive with ${\kappa }_R=\frac{1}{440}\left[s^{-2}\right]$

$$\left(\begin{array}{l}\frac{\partial f^R}{\partial x}\\ \frac{\partial f^R}{\partial y}\end{array}\right)=-{\kappa }_R\left(\frac{109}{{\delta }_o}-288+36{\delta }_o\right)\left(\begin{array}{l}{x}_o-x_t\\ y_o-y_t\end{array}\right).$$E33

The repulsive field is a negative function that works within range $0\le {\delta }_o\le 8$,

$$\underset{{\delta }_o\to 8}{lim}{f}^R\left({\delta }_o\right)=0;\text{and}\underset{{\delta }_o\to 0}{lim}{f}^R\left({\delta }_o\right)=5.$$E34

The direction fields produced are shown in Figure 5, where for both cases the coordinates origin represents either goal or obstacle locations.

Figure 5.

Acceleration fields w.r.t. variations along the plane xy. (a) Goals attractive motion generation. (b) Obstacles avoidance motion generation.

Moreover, when neither goals of interest nor obstacles are within the observation field of the robot, it must keep navigating along a prior route plan. The routing plan is map comprised of a sequence of Cartesian points ${\mathbf{g}}_k\in {\mathbb{R}}^2$, ${\mathbf{g}}_k={\left(x,y\right)}^{\top }$.

When the robot accomplishes either $f^A$ or $f^R$, it continues toward the following route point, expressed in terms of unit vectors:

$${\mathbf{a}}_t^o=a^o\frac{{\mathbf{g}}_{k+1}-{\mathbf{g}}_k}{‖{\mathbf{g}}_{k+1}-{\mathbf{g}}_k‖},$$E35

where a^o is an ideal or averaged acceleration toward the next route point g₂ from g₁. Therefore, the total path planner mode is given by Proposition 3.1.

Proposition 3.1 Total mission path planning. The total path planning model subjected to adaptive environmental changes is proposed as a vector fields sum:

$${\mathbf{a}}_t={\mathbf{a}}_t^o+\sum _g\left[{\kappa }_A\left(\frac{3{\delta }_g}{2}-\frac{6}{{\delta }_g}-12\right)\left(\frac{x_g-x_t}{y_g-y_t}\right)+\sum _o{\kappa }_R\left(288-\frac{109}{{\delta }_o}-36{\delta }_o\right)\left(\begin{array}{l}{x}_0-x_t\\ y_o-y_t\end{array}\right)\right]$$E36

and the robot’s instantaneous navigational total velocity ${\mathbf{v}}_T\in {\mathbb{R}}^2$, ${\mathbf{v}}_T={\left(x_T,{\dot{y}}_T\right)}^{\top }$ is obtained by

$${\mathbf{v}}_T={\int }_{t_1}^{t_2}{\mathbf{a}}_t\mathrm{d}t.$$E37

Thus, to automatically limit the speed until the robot’s maximal velocity v_max, the real physical velocity v_ph is constrained by the final planning motion

$$v_{\mathit{ph}}\left(\frac{‖{\mathbf{v}}_T‖}{v_{\mathit{max}}}\right)=\left\{\begin{array}{ll}\frac{‖{\mathbf{v}}_T‖}{v_{\mathit{max}}},& \frac{‖{\mathbf{v}}_T‖}{v^{\mathit{max}}}<1\\ v_{\mathit{max}},& \frac{‖{\mathbf{v}}_T‖}{v^{\mathit{max}}}\ge 1\end{array}\right.$$E38

Figure 6 shows simulation results of the total navigation planner.

Figure 6.

Vector fields path generation. (a) High-density obstacles yielding robot’s repulsion. (b) Obstacles repulsive field. (c) High density $‖{\mathbf{f}}^R‖$. (d) Attractive goals along a route.

4. Path tracking

This section deduces an algebraic trajectory tracking linearized controller based on the kinematics geometry of the reference path: linear and angular velocities. Considering from previous Section 3, let v_ph be called the reference velocity that is going to be tracked. Likewise, let (x_1,2, y_1,2) be the next local planning coordinates to be reached (from location 1 to location 2). Hence, let us define $x_{1,2}\doteq x_2-x_1$ and $y_{1,2}\doteq y_2-y_1$. Such that ${\dot{s}}_{1,2}\left[\mathrm{m}/\mathrm{s}\right]$ is a segment of the planning speed. Hence, let us obtain the first-order derivative,

$${\dot{s}}_{1,2}=\frac{d}{\mathit{dt}}\sqrt[2]{x_{1,2}^2+y_{1,2}^2}=\frac{x_{1,2}{\dot{x}}_{1,2}+y_{1,2}{\dot{y}}_{1,2}}{\sqrt[2]{x_{1,2}^2+y_{1,2}^2}}.$$E39

Similarly, by obtaining the desired robot’s orientation ${\theta }_{1,2}$, its first-order derivative w.r.t. time is

$${\dot{\theta }}_{1,2}=\frac{d}{\mathit{dt}}\left(arctan\left(\frac{y_{1,2}}{x_{1,2}}\right)\right)=\frac{x_{1,2}{\dot{y}}_{1,2}-y_{1,2}{\dot{x}}_{1,2}}{x_{1,2}^2+y_{1,2}^2}.$$E40

It follows that, the tracking control vector is ${\dot{\mathbf{u}}}_t\in {\mathbb{R}}^2$, such that ${\dot{\mathbf{u}}}_t={\left({\dot{s}}_{1,2},{\dot{\theta }}_{1,2}\right)}^{\top }$. Therefore, by stating (39) and (40) as a system of linear equations, the first-order derivatives must simultaneously be solved:

$$\begin{array}{ll}\frac{x_{1,2}}{s_{1,2}}{\dot{x}}_{1,2}+\frac{y_{1,2}}{s_{1,2}}{\dot{y}}_{1,2}& =\dot{s}\\ -\frac{y_{1,2}}{s_{1,2}^2}{\dot{x}}_{1,2}+\frac{x_{1,2}}{s_{1,2}^2}{\dot{y}}_{1,2}& =\dot{\theta }\end{array},$$E41

thus, by factorizing both, the common term $1/s_{1,2}$ and the first-order derivatives, the matrix form of the forward tracking law is

$$\dot{\mathbf{u}}=\frac{1}{s_{1,2}}\left(\begin{array}{cc}{x}_{1,2}& y_{1,2}\\ -\frac{y_{1,2}}{s_{1,2}}& \frac{x_{1,2}}{s_{1,2}}\end{array}\right)\cdot \left(\begin{array}{l}{\dot{x}}_{1,2}\\ {\dot{y}}_{1,2}\end{array}\right).$$E42

Likewise, as the inverse tracking law is of our interest, it is inversely dropped off

$$\left(\begin{array}{l}{\dot{x}}_{1,2}\\ y_{1,2}\end{array}\right)=s_{1,2}{\left(\begin{array}{cc}{x}_{1,2}& y_{1,2}\\ -\frac{y_{1,2}}{s_{1,2}}& \frac{x_{1,2}}{s_{1,2}}\end{array}\right)}^{-1}\cdot \dot{\mathbf{u}}.$$E43

For notation simplicity, let us redefine ${\dot{\zeta }}_t\doteq {\left({\dot{x}}_{1,2},{\dot{y}}_{1,2}\right)}^{\top }$ and the time-variant control matrix K_t as

$${\mathbf{K}}_t=\left(\begin{array}{cc}{x}_{1,2}& y_{1,2}\\ -\frac{y_{1,2}}{s_{1,2}}& \frac{x_{1,2}}{s_{1,2}}\end{array}\right),$$E44

thus from Section 3, let u^ref be the global reference planning model during time segment $t_{1,2}\doteq t_2-t_1$ such that,

$${\mathit{u}}^{\mathit{ref}}=\left(\begin{array}{c}{v}_{\mathit{ph}}\cdot t_{1,2}\\ arctan\left(\frac{{\dot{y}}_T}{{\dot{x}}_T}\right)\end{array}\right).$$E45

Therefore, the recursive path tracking control law is stated as

$$\begin{array}{ll}1:& {\zeta }_{t+1}={\zeta }_t+s_{1,2}{\mathbf{K}}_t^{-1}\cdot \left({\mathit{u}}^{\mathit{ref}}-{\widehat{\mathbf{u}}}_t\right)\\ 2:& {\mathbf{u}}_{t+1}={\mathbf{u}}_t+\frac{1}{s_{1,2}}{\mathbf{K}}_t\cdot \left({\zeta }_{t+1}-{\widehat{\zeta }}_t\right)\end{array}$$E46

where ${\widehat{\mathbf{u}}}_t$ is the instantaneous sensors observation of both components, displacement $\widehat{s}$ and yaw $\widehat{\theta }$. The prediction control speed vector ${\zeta }_{t+1}$ is the next desired local reference in line 2, while ${\widehat{\zeta }}_t$ is the robot’s Cartesian speed observer vector. Finally, u_t + 1 is the control vector global prediction (Table 1).

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
l [m]	0.03	a [m]	0.25	${\kappa }_A\left[s^2\right]$	3/75	Ix [kg^2]	0.12
d [m]	0.15	bl [m]	0.4	${\kappa }_R\left[s^{-2}\right]$	1/440	Iy [kg^2]	0.12
ea [m]	0.08	${\mathrm{\ell }}_b\left[m\right]$	0.75	a^o [m/s^2]	0.5	Iz [kg^2]	0.13
r [m]	0.15	m [kg]	0.76	rw [m]	0.1	${\upsilon }_{\mathit{max}}\left[m/s\right]$	1.5

Table 1.

Simulation parameters deployed with high-density range scanner Hokuyo URG 04LX.

^a

High stiffness non-rigid bar.

5. Conclusion

The mechanical design of the biped’s lower-limb mechanical structure was configured as a double-crank four-bar linkage with passive-allowed motions. Motion planning began from determining the limbs’ linkage positions causing the robot’s height and pitch varying overtime producing unbalanced motions. Inferring balancing velocities to yield robot’s vertical balance was possible and worked stable. The proposed balancing rolling condition was analyzed throughout its total energy model as a Lyapunov candidate function resulting stable in three criteria: V(0) = 0, v(x) = 0, and $\dot{V}\left(x\right)=0$. The proposed navigational approach allowed human-user to design short range-limited trajectories by cubic polynomials obtained from four empirical coordinates distance-acceleration by Lagrange interpolation. Both polynomial planning models have initial conditions $a_0\left(t_0\right)=0\left[m/s^2\right]$. Thus, the robot did not require an infinite acceleration to reach a desired speed at t₀. Physically, the model is applied to any motion in equilibrium. For goal attraction the maximal acceleration is reached at the 55% of the distance, subsequently decreases monotonically until the goal position.

The navigational general planning model is a set of partial derivatives model combined, allowing dynamic local planning among multiple obstacles, goals, and routes. The navigational general planning model worked as the reference model for the tracking control system. It consisted of a set of time-varying linear equations, with recursive feedback showing suitable performance. The work was implemented in simulation and coded in C++ under Linux. Future improvements will consider the inclusion of kinetic models and dynamic forces fitted to the proposed planer.

This appendix provides the algebraic deduction of the limb’s Cartesian motion model described in Section 2.

Let ${\mathbf{P}}_1={\left(x_1,y_1\right)}^{\top }$ be the Cartesian point of passive joint between links b and $\mathrm{\ell }$, obtained by

$$\left(\begin{array}{l}{x}_1\\ y_1\end{array}\right)=r\left(\begin{array}{l}cos\left(\varphi \right)\\ sin\left(\varphi \right)\end{array}\right).$$E47

Likewise, let ${\mathbf{P}}_2={\left(x_2,y_2\right)}^{\top }$ be the coordinate of passive joint between links r and $\mathrm{\ell }$, expressed by:

$$\left(\begin{array}{l}{x}_2\\ y_2\end{array}\right)=r\left(\begin{array}{l}cos\left(\varphi \right)\\ sin\left(\varphi \right)\end{array}\right)+\left(\begin{array}{l}\sqrt{l^2-{\left(y_1-y_2\right)}^2}\\ \sqrt{l^2-{\left(x_1-x_2\right)}^2}\end{array}\right).$$E48

Both expressions (47) and (48) are described in terms of $\varphi$, but can similarly be described in terms of the angle $\theta$ by

$$\left(\begin{array}{l}{x}_2\\ y_2\end{array}\right)=d\left(\begin{array}{l}cos\left(\theta \right)\\ sin\left(\theta \right)\end{array}\right)+\left(\begin{array}{l}\mathrm{\Delta }x\\ \mathrm{\Delta }y\end{array}\right),$$E49

where the distances $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ separate the joints phi and theta along the chassis, see Figure 2(a). By algebraically treating the Cartesian components separately, let us equal the x components of (48) and (49) to form the equation

$$dcos\left(\theta \right)+\mathrm{\Delta }x=rcos\left(\varphi \right)+\sqrt{l^2-{\left(y_1-y_2\right)}^2},$$E50

dropping off the squared root term of (50),

$$\sqrt{l^2-{\left(y_1-y_2\right)}^2}=dcos\left(\theta \right)+\mathrm{\Delta }x-rcos\left(\varphi \right)$$E51

and squaring both sides of the equality, the following is obtained,

$$l^2-{\left(y_1-y_2\right)}^2={\left[dcos\left(\theta \right)+\mathrm{\Delta }x-rcos\left(\varphi \right)\right]}^2$$E52

and algebraically expanding the squared binomials and substituting y₁ and y₂ from expression (47) and (49) respectively into (52),

$$\begin{array}{l}{d}^2{cos}^2\left(\theta \right)+2\mathrm{\Delta }\mathit{xd}cos\left(\theta \right)-2\mathit{rd}cos\left(\theta \right)cos\left(\varphi \right)+{\left(\mathrm{\Delta }x\right)}^2\\ -2\mathrm{\Delta }\mathit{xr}cos\left(\varphi \right)+r^2{cos}^2\left(\varphi \right)\\ =l^2-r^2{sin}^2\left(\varphi \right)+2\mathit{rd}sin\left(\theta \right)sin\left(\varphi \right)+2\mathrm{\Delta }\mathit{yr}sin\left(\varphi \right)\\ -d^2{sin}^2\left(\theta \right)-2\mathrm{\Delta }\mathit{yd}sin\left(\theta \right)-{\left(\mathrm{\Delta }y\right)}^2\end{array}$$E53

and organizing trigonometric terms by their degree, Eq. (53) becomes:

$$\begin{array}{l}2\mathit{rd}cos\left(\theta \right)cos\left(\varphi \right)+2\mathit{rd}sin\left(\theta \right)sin\left(\varphi \right)\\ =d^2{cos}^2\left(\theta \right)+d^2{sin}^2\left(\theta \right)+{\left(\mathrm{\Delta }x\right)}^2+{\left(\mathrm{\Delta }y\right)}^2\\ +r^2{cos}^2\left(\varphi \right)+r^2{sin}^2\left(\varphi \right)+2\mathrm{\Delta }\mathit{xd}cos\left(\theta \right)-2\mathrm{\Delta }\mathit{xr}cos\left(\varphi \right)\\ +2\mathrm{\Delta }\mathit{yd}sin\left(\theta \right)-2\mathrm{\Delta }\mathit{yr}sin\left(\varphi \right).\end{array}$$E54

By factorizing the common terms of (47) and exchanging terms with suitable identities such as ${cos}^2\left(\alpha \right)+{sin}^2\left(\alpha \right)=1$ and $c^2={\left(\mathrm{\Delta }x\right)}^2+{\left(\mathrm{\Delta }y\right)}^2$ in order to obtain Eq. (55)

$$\begin{array}{l}cos\left(\theta \right)cos\left(\varphi \right)+sin\left(\theta \right)sin\left(\varphi \right)=\\ \frac{r^2+d^2+c^2-l^2-2r\left[\mathrm{\Delta }xcos\varphi +\mathrm{\Delta }ysin\left(\varphi \right)\right]+2d\mathrm{\Delta }xcos\left(\theta \right)+2d\mathrm{\Delta }ysin\left(\theta \right)}{2\mathit{rd}}\end{array}$$E55

In order to simplify the main expression, some constant terms are rewritten as

$$K_1=\frac{r^2+d^2+c^2-l^2}{2\mathit{rd}},K_2-\frac{1}{d},K_3=\frac{\mathrm{\Delta }x}{r},K_4=\frac{\mathrm{\Delta }y}{r}.$$E56

Therefore, the following simplified algebraic expression is deduced:

$$\begin{array}{l}cos\left(\theta \right)cos\left(\varphi \right)+sin\left(\theta \right)sin\left(\varphi \right)=K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]\\ +K_{3}cos\left(\theta \right)+K_{4}sin\left(\theta \right).\end{array}$$E57

The trigonometric identity forms $sin\left(\theta \right)=\frac{2tan\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}$ and $cos\left(\theta \right)=\frac{1-{tan}^2\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}$ are substituted into (57) to produce (58),

$$\begin{array}{l}cos\left(\varphi \right)\left[\frac{1-{tan}^2\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}\right]+sin\left(\varphi \right)\left[\frac{2tan\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}\right]=K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]\\ +K_3\left[\frac{1-{tan}^2\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}\right]+K_4\left[\frac{2tan\left(\frac{\theta }{2}\right)}{1+{tan}^2\left(\frac{\theta }{2}\right)}\right]\end{array}$$E58

and by multiplying both sides of the equality (57) by common denominator $1+{tan}^2\left(\frac{\theta }{2}\right)$ and algebraically simplifying:

$$\begin{array}{l}cos\left(\varphi \right)-cos\left(\varphi \right){tan}^2\left(\frac{\theta }{2}\right)+2sin\left(\varphi \right)tan\left(\frac{\theta }{2}\right)=K_1+K_1{tan}^2\left(\frac{\theta }{2}\right)\\ +K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]{tan}^2\left(\frac{\theta }{2}\right)\\ +K_3-K_3{tan}^2\left(\frac{\theta }{2}\right)+2K_{4}tan\left(\frac{\theta }{2}\right).\end{array}$$E59

Hereafter, to find a root for θ as independent variable, common terms are factorized and Eq. (59) is equated to zero to obtain

$$\begin{array}{l}{tan}^2\left(\frac{\theta }{2}\right)\left[K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]-K_3+cos\left(\varphi \right)\right]\\ +tan\left(\frac{\theta }{2}\right)\left[2K_4-2sin\left(\varphi \right)\right]+K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]\\ +K_3-cos\left(\varphi \right)=0,\end{array}$$E60

in order to reduce expression complexity, the following Definitions are stated:

Definition 4.1 [term P]. Let P be defined as a Cartesian horizontal decreasing segment by expression:

$$P=K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]-K_3+cos\left(\varphi \right).$$E61

Definition 4.2 [term Q]. Let Q be defined as a Cartesian vertical segment that is expressed by:

$$Q=2K_4-2sin\left(\varphi \right)$$E62

and

Definition 4.3 [term P]. Let R be defined as a Cartesian horizontal increasing segment by expression:

$$R=K_1+K_2\left[\mathrm{\Delta }xcos\left(\varphi \right)+\mathrm{\Delta }ysin\left(\varphi \right)\right]+K_3-cos\left(\varphi \right)$$E63

Therefore, by substituting the terms of Definitions 4.1–4.3 into Eq. (60), the following quadratic form is deduced,

$$P{tan}^2\left(\frac{\theta }{2}\right)+Qtan\left(\frac{\theta }{2}\right)+R=0.$$E64

By analytically solving (64), which is a second-degree equation using the general form, a real solution for θ is possible, thus

$$\theta =2arctan\left(\frac{-Q\pm \sqrt{Q^2-4\mathit{PR}}}{2P}\right)$$E65