Simulation parameters deployed with high-density range scanner Hokuyo URG 04LX.
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.
- path planning
- motion control
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 . 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  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  modeled a wheeled bipedal robot with analytic solutions of closed-form expressions in kinematic control loops. The work  reported a self-balancing two-wheeled robot with a manipulator on-board, using auto-balancing system to maintain force equilibrium. The work  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 , 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 , a navigational two-wheeled self-balancing robot control using a PD-PI controller based on the Kalman filter algorithm was reported. Similarly,  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 . Moreover,  conducted a study using model predictive control for trajectory tracking of an inverted-pendulum wheeled robot. The work  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  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.
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 , 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.
The expressions provided in Proposition 2.1 are obtained from deductions in Appendix A. The passive joint point mutually depends on the analytic model of the joint located at to describe a rotary planar motion, proposed by
From Eq. (56) deduced in Appendix A, there are two possible solutions for
Furthermore, the wheels position is tracked by
Hence, in accordance to Definition 2.1, it follows that the robot’s falling speed vf due to vertical unbalance is
Taking into account that the robot is a dual differential drive kinematic structure, where vR and vL are the right-sided and left-sided velocities, respectively
Hence, the robot’s angular velocity in terms of its differential speed
The biped’s falling angle and angular falling speed are and , respectively. Let
which makes the robot’s pitch turn around the wheel’s center. Unbalancing velocity
The wheel’s axis point moves at balancing speed
Therefore, the balancing condition is described by Definition 2.2:
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,
Thus, let us redefine the state variables as , , . For an stability analysis for the balancing case when , and equilibrium condition occurs,
and dropping off the highest-order derivative of the system
The system total energy (kinetic plus potential) is a positive function, which is used as a Lyapunov candidate function, where the robot’s translation motion for equilibrium is
and substituting ,
For and finding out that v(x) and fulfill . In addition, assuming that , then . Assuming that is a continuous differentiable function. Finally, the following Lyapunov criterion is satisfied (20),
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:
where the human-user establishes a desired numerical acceleration
It follows that, Definition 3.1 establishes the acceleration path planning model toward a goal of interest.
Similarly, Definition 3.2 establishes the acceleration path planning model that avoids obstacles zones.
Figure 4(a) shows the robot’s instantaneous acceleration toward a goal of interest. The attraction acceleration starts when the goal-robot distance . The planner allows start from 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 . This avoidance planner is faster than to increase confidence against obstacles.
Therefore, extending to two-dimension Cartesian space, let us deduce the following algebraic process for the goal-attraction planner :
as well as
Similarly, for the robot’s acceleration to avoid obstacles, let us develop the following , by substituting the functional form of into the gradient function
subsequently by applying the gradient operator,
Thus, by substituting terms in both local planners, the general attractive motion planning is a negative function with amplitude coefficient is represented by
The attractive field is a positive function working within limits , thus
and the repulsive with
The repulsive field is a negative function that works within range ,
The direction fields produced are shown in Figure 5, where for both cases the coordinates origin represents either goal or obstacle locations.
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 , .
When the robot accomplishes either or , it continues toward the following route point, expressed in terms of unit vectors:
and the robot’s instantaneous navigational total velocity , is obtained by
Thus, to automatically limit the speed until the robot’s maximal velocity
Figure 6 shows simulation results of the total navigation planner.
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
Similarly, by obtaining the desired robot’s orientation , its first-order derivative w.r.t. time is
thus, by factorizing both, the common term and the first-order derivatives, the matrix form of the forward tracking law is
Likewise, as the inverse tracking law is of our interest, it is inversely dropped off
For notation simplicity, let us redefine and the time-variant control matrix
thus from Section 3, let
Therefore, the recursive path tracking control law is stated as
where is the instantaneous sensors observation of both components, displacement and yaw . The prediction control speed vector is the next desired local reference in line 2, while is the robot’s Cartesian speed observer vector. Finally,
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:
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 be the Cartesian point of passive joint between links b and , obtained by
Likewise, let be the coordinate of passive joint between links
where the distances and separate the joints
dropping off the squared root term of (50),
and squaring both sides of the equality, the following is obtained,
and organizing trigonometric terms by their degree, Eq. (53) becomes:
In order to simplify the main expression, some constant terms are rewritten as
Therefore, the following simplified algebraic expression is deduced:
and by multiplying both sides of the equality (57) by common denominator and algebraically simplifying:
Hereafter, to find a root for
in order to reduce expression complexity, the following Definitions are stated:
Therefore, by substituting the terms of Definitions 4.1–4.3 into Eq. (60), the following quadratic form is deduced,
By analytically solving (64), which is a second-degree equation using the general form, a real solution for