Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6725427, 15 pages

https://doi.org/10.1155/2017/6725427

## Constrained Quadratic Programming and Neurodynamics-Based Solver for Energy Optimization of Biped Walking Robots

^{1}Department of Electronic Engineering, Shunde Polytechnic, Foshan, Guangdong 528300, China^{2}School of Automation, Xi’an University of Posts and Telecommunications, Xi’an 710121, China^{3}College of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Liyang Wang

Received 17 February 2017; Revised 16 July 2017; Accepted 3 August 2017; Published 17 September 2017

Academic Editor: Dan Simon

Copyright © 2017 Liyang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The application of biped robots is always trapped by their high energy consumption. This paper makes a contribution by optimizing the joint torques to decrease the energy consumption without changing the biped gaits. In this work, a constrained quadratic programming (QP) problem for energy optimization is formulated. A neurodynamics-based solver is presented to solve the QP problem. Differing from the existing literatures, the proposed neurodynamics-based energy optimization (NEO) strategy minimizes the energy consumption and guarantees the following three important constraints simultaneously: (i) the force-moment equilibrium equation of biped robots, (ii) frictions applied by each leg on the ground to hold the biped robot without slippage and tipping over, and (iii) physical limits of the motors. Simulations demonstrate that the proposed strategy is effective for energy-efficient biped walking.

#### 1. Introduction

Biped robots receive a lot of attention from the scientists and engineers these years [1–10]. Compared with wheeled driven robots and crawler robots, biped robots have anthropomorphic adaptability to unstructured environments with relative less energy consumption. However, Honda’s ASIMO, perhaps representative of humanoid robots, uses at least 10 times the energy (scaled as dimensionless specific cost of transport) of a typical human [11]. Energy consumption of the existing biped robots is much more than that of human being.

Two main kinds of approaches have been presented to optimize the energy efficiency for legged robots: the first type focuses on the improvements of mechanical device [11–14] and the second type is devoted to better optimization algorithm [15–19].

A well-known improvement of biped mechanical device is brought by “semipassive walkers.” Numerous very successful “semipassive walkers” have been built [11], such as the Cornell biped, the Delft biped, and the MIT learning biped robots. These robots are built based on passive dynamics, with small active power sources substituted for gravity, which can walk on level ground. However, uneven ground leads to falls of the Cornell biped robot and floor irregularities lead to falls of the Delft biped robot. Although the MIT learning biped robot continually learns and adapts to the terrain as it walks, its energy consumption is especially high.

From another aspect, several auxiliary devices are designed to decrease energy consumption of legged robots. In [12], an elastic load suspension mechanism was developed and utilized on a hexapod robot to increase the energy efficiency of legged robot locomotion. The experiments demonstrate that the robot with an elastically suspended load consumed up to 24% less power than with a rigidly attached load. In [13], a flexible shoe system was proposed for biped robots to optimize energy consumption of the lateral plane motion. The shoe system can absorb the kinetic energy of the robot, which is confirmed by simulations and experiments. In [14], the energy efficiency of a humanoid robot with spinal motion was investigated. Simulations show that, with the additional degrees of freedom (DoF) in the torso, the robot requires 26.5% less energy than its rigid-torso counterpart to complete the same walking.

Differing from various mechanical devices for decreasing the energy consumption of legged robots, optimization algorithms can be used more widely because this kind of approach is applicable to all the robots with actuators. Motivated by trunk rotations of human being, an algorithm based on fuzzy logic and iterative optimization [15] provides a remarkable descend rate of energy consumption for biped robots. In [16], an online gait synthesis algorithm compromising waking stability and energy efficiency is proposed and the parameters are optimized for a given travel distance minimizing the energy consumed by the actuators. In [17], a reference generation algorithm was proposed for biped robots based on linear inverted pendulum model. Simulation results suggest that the proposed moving zero-moment-point (ZMP) references are more energy-efficient than the ones with fixed ZMP under the supporting foot. However, when decreasing the energy consumption of biped robots, the reported works neglect the inevitable presence of some inside constraints, including the reaction forces from the ground, frictional force between the feet and the ground, and some limitations of the biped joints.

On the other hand, some recent works have been done in modeling, measuring, estimating, and rejecting the undesired force introduced to the dynamics of robots. Fakoorian et al. [18] designed a continuous-time extended Kalman filter (EKF) and a continuous-time unscented Kalman filter (UKF) to estimate not only the states of the robot system but also the ground reaction forces that act on the foot. Moosavi et al. [19] designed a nonlinear controller for second-order nonlinear uncertain dynamical systems. The results demonstrate that the proposed method is a partly model-free controller that works well in certain and partly uncertain system. Differing from the existing literatures, the proposed neurodynamics-based energy optimization (NEO) strategy minimizes the energy consumption and guarantees the following three important constraints simultaneously: (i) the force-moment equilibrium equation of biped robots, (ii) frictions applied by each leg on the ground to hold the biped robot without slippage and tipping over, and (iii) Physical limits of the motors.

In this work, in order to provide optimal controller to meet all the above-mentioned constraints, a QP problem is formulated to deal with the constrained optimization problem. Utilizing the dynamics model, an objective function based on energy-efficiency optimization can be computed so that the control input can be produced within system constraints. Furthermore, the optimized control outputs need to be available as the robot moves on at each step. Therefore, online implementation is the next fundamental issue in the energy efficiency optimization. Whether the energy optimum control can be implemented successfully is dependent on the efficiency of online optimization.

To solve the proposed constrained optimization problem in real time, neurodynamics-based QP solvers are considered because of their sufficient online computational power [20–22]. Several optimal controls for robots have been realized using neural network optimization [23–25]. In this paper, considering the physical constraints on biped motions, a novel neurodynamics-based energy-efficiency optimization controller is developed. The strategy for optimizing the energy efficiency is formulated as a constrained QP problem, which can be solved online using a primal-dual neural network (PDNN). Simulation studies are presented to illustrate the feasibility of the proposed strategy.

Main contributions of this work could be summarized as follows:(i)A new complete model for energy optimization of biped walking robots is built. Several necessary physical constraints for the biped system are formulated in the optimization controller, which provides a more practical objective for the energy optimization of biped robots.(ii)A neurodynamics-based solver is presented to handle the proposed optimization problem. To the best of our knowledge, no works on biped walking using energy optimum control based on neural network have been reported before.

The organization of this paper is as follows. In Section 2, the background about the energy consumption of biped walking robots is presented. A QP problem for the energy optimization is formulated in Section 3. Section 4 proposed a neurodynamics-based solution for the proposed QP problem to realize the energy optimization. Section 5 presented the diagram of the neurodynamics-based energy optimization control system. Simulation results are provided in Section 6, followed by the conclusions in Section 7.

#### 2. Energy Consumption of a Biped Walking Robot

The main idea is to optimize the energy consumption of a biped robot walking on horizontal ground. For optimization purposes, we consider the following assumptions.

*Assumption 1. *We assume that the robot walks on practicable horizontal terrain. Therefore, there are no forbidden zones, and every foot can be placed on the ground and provide support to the robot.

*Assumption 2. *We assume that all the motors in the biped joints have the same configurations. The robot’s actuators are based on DC electric motors (electrical motors cannot store the gains of negative energy) [26, 27].

*Assumption 3. *Consider a biped robot with point contacts between the ground and each foot. Assume that each foot contacts the ground with Coulomb friction.

The energy consumption of a biped robot is given by the sum of the energy consumed in every joint plus the energy consumed by electronic equipment (computers, drivers, analog I/O, etc.). The contribution of the latter is out of the scope of this work.

If a joint is driven by an electrical motor, the consumed energy is given bywhere is the voltage and is the current through the motor. As a normal practice in modeling DC motors for robotic joints [27], the following electrical model of a motor is considered here:where represents the resistance of the motor and denotes the rotation speed of the motor. is the back electromotive constant, denotes the torque constant, and represents the motor torque. As a first approximation to the model, we consider the rotor inductance, , to be null. Note that in standard international unit the value of is the value of ; that is, (Nm/A) (Vs).

Substituting (2) into (1), we obtain

According to Assumption 2, the energy expended by the motor of joint of leg during time can be given bywhere denotes the serial number of the legs and is the serial number of the joints/motors in one leg. The function is defined asTaking into consideration all joints in the biped robot, the total energy consumption becomesConsidering the practical differences between the rotation speed and torques in joint and motor of leg , joint gear ratio and the joint gear’s mechanical efficiency are introduced here. Thus, and , where and represent the speed and torques in joint of leg , while and represent those in motor of leg . Therefore, (6) can be rewritten asThe main goal of this work is to minimize the expended energy given by (7). Then, the objective function can be presented as

In the next section, the energy-efficiency optimization problem will be formulated as a QP problem, which can be solved online using neurodynamics.

*Remark 4. *Load plays a crucial role in the application of the robot. In [28], the combination of neural network (NN), proportional derivative (PD), and robust controller is used for determining the maximum load-carrying capacity (MLCC) of articulated robots, subject to both actuator and end-effector deflection constraints. The proposed technique is then applied to articulated robots, and MLCC is obtained for a given trajectory. Literature [29] provides accurate estimates of the robot load inertial parameters, and accurate actuator torques predictions, both of which are essential for the acceptance of the results in an industrial environment. The key element to the success of this work is the comprehensiveness of the applied model, which includes, besides the dynamics resulting from the robot load and motor inertia, the coupling between the actuator torques, the mechanical losses in the motors, and the efficiency of the transmissions.

From the literatures, we know that load is not the only factor that demands the torque. In this work, the weight of the robot is considered as the load.

#### 3. Formulate a QP Problem for Energy-Efficient Biped Robots

The objective function shown in (8) can be expressed as the following optimization problem:where. denotes that the elements in the matrix are equal to the bigger elements between in the matrix and in the matrix . The problem shown in (9) should guarantee three necessary constraints:(i)The force-moment equilibrium equation of biped robots(ii)Frictions applied by each leg on the ground to hold the biped robot without slippage and tipping over(iii)Physical limits of the motors

Let us begin with the force-moment equilibrium equation for the whole biped system, which can be written aswhere is a vector of the ground reaction forces acting on the left foot and the right foot.where denotes the position of leg referring to basic coordinate frame. contains the forces and moments acting on the robot’s center of gravity (COG).

Biped dynamics provide the relevance between ground reaction forces acting on the feet and the joint torques of the legs. Consider the th leg of a biped robot; the dynamics of the system can be expressed in the vector-matrix form aswhere is a joint position vector of the th leg; for the biped robot*. * is the inertia matrix of the leg, is a vector of centrifugal and Coriolis terms, is a vector of gravity terms, is the vector of joint torques, is the vector of ground reaction forces of the th leg, and is the Jacobian matrix from the joints of the th leg to the basic coordinate space. Equation (13) can be rewritten asReplacing in (11) with (14), the force-moment equilibrium equation (11) can be written aswhere is an equilibrium matrix for the torque andEquation (15) is the first constraint describing the force-moment equilibrium equation of biped robots.

On the other hand, considering the friction applied by the leg on the ground, the second constraint can be written aswhere is the coefficient matrix of the friction constraint for the feet. where is the Coulomb friction coefficient between the feet and the touched ground.

The third constraint is about the physical limits of the motors, which has the form of . and are known torque vectors according to the motors in the robot: and .

From the above, the energy-efficiency optimization problem can be formulated as the following QP problem:where the definitions of the symbols can be found in (9), (15), and (17).

As we can see, now the strategy for minimizing the energy consumption of biped robots is finally formulated as a constrained QP problem (see (19)–(22)). In the next section, we are going to solve this QP problem using neurodynamics.

#### 4. Solve the QP Problem for Energy Optimization

First of all, the QP problem described in (19)–(22) needs to be converted to a LVI problem. For this purpose, and are defined as the dual decision vectors corresponding to the equality constraint (20) and the inequality constraint (21), respectively. As a result, the primal-dual decision vector and its upper/lower bounds are defined as follows:where elements in are defined sufficiently positive to represent infinity for any for the convenience of simulation and hardware implementation of the operation. The convex set made by primal-dual decision vector is presented as . Then the QP problem in (19)–(22) is equivalent to the following LVI problem:That is to say, we need to find a vector with a coefficient matrix and a vector as follows:As a summary of this step, a theorem and the corresponding proof are given below.

Theorem 5 (see [21] (LVI formulation)). *The QP problem in (19)–(22) is equivalent to the following LVI problem, that is, to find a vector such that , where coefficients , , and are defined in (24) and (23), respectively.*

*Proof. *From [21], we know that the Lagrangian dual problem of the QP problem in (19)–(22) is to maximize over and , whereNote that function is convex with given , and . Therefore, the necessary and sufficient condition for a minimum isThus, the dual problem of (19)–(22) can be derived asMultiplying both sides of (27) by , we have Then the dual form can be rewritten asTo obtain the optimum of the primal problem (see (19)–(22)) and its dual problem (see (30)), the necessary and sufficient conditions are as follows: Primal feasibility: Dual feasibility: Complementarity:To simplify the above necessary and sufficient conditions, by defining and considering (35)-(36), dual feasibility constraint (33) becomeswhich is equal to the following LVI problem: to find an optimum such thatwhere .

In addition, by defining , we have the following LVI for the equality constraint (31): to find such thatSimilarly, by defining , we have the following LVI for the inequality constraint (32) and the equality constraint (34): to find such thatBy taking the Cartesian product of the three sets , , and defined above, we formulate the convex set presented in (24) as follows:In consequence, LVIs (38)–(40) can be combined into an LVI in the matrix form, that is, to find an optimum such that , andContrasting the definitions of in (23) and and in (25), the above LVIs (see (42)) are exactly in the compact form (see (25)) as the equivalence of QP problem in (19)–(22). The proof is then completed.

On the other hand, from [20], it is known that LVI in (25) is equivalent to the following system of piecewise linear equations:where is a piecewise linear projection operator with the th projection element of being defined asTo solve linear projection equation (43), the dual dynamical system design approach presented in [20] is adopted to build a dynamical system. Considering the asymmetric , a primal-dual neural network is developed as follows to solve (43):where is a design parameter used to scale the convergence rate of the system. The convergence rate is improved when the value of is increased. Furthermore, by replacing with , the dynamical system (45) can be rewritten as follows:where is a positive design parameter for the convergence rate of the PDNN (see (46)). About the convergence analysis of the above-mentioned primal-dual dynamical solver, there is a theorem presented as follows.

Theorem 6 (see [22]). *Starting from any initial state, the state vector of primal-dual dynamical system (45) is convergent to an equilibrium point , of which the first elements constitute the optimal solution to the quadratic programming problem in (19)–(22). Moreover, the exponential convergence can be achieved if there exists a constant such that*

*Remark 7. *As a primal-dual dynamical solver, primal-dual neural networks can efficiently generate exact optimal solutions [21]. Furthermore, compared to dual neural networks, the LVI- (linear variational inequality-) based primal-dual neural networks do not entail any matrix inversion online, so that they can reduce the computational time greatly [20]. Therefore, a LVI-based primal-dual neural network is applied to perform the minimization of the energy consumption with equality and inequality constraints for biped robots in this work.

#### 5. Neurodynamics-Based Energy Optimization Control

Over the past two decades, much progress has been made in the control of robot systems based on various control techniques. In [30], two novel robust adaptive PID control schemes are proposed to solve the strong nonlinearity and coupling problems in robot manipulator control. In [31], a support vector regression-based control system is proposed to learn the external disturbances and increase the zero-moment-point stability margin of humanoid robots. In [32], a kinematics open-loop control system of hexapod robot with an embedded digital signal controller is proposed. In [33], an open-loop controller is implemented on field programmable gate array (FPGA) of the robot. Experimental results reveal the viability of the proposed controller.

On the other hand, many mature techniques have been used to track the planned trajectories with a certain control strategy. In [34], a trajectory generator and a joint trajectory tracking controller are designed. The proposed combination is successfully implemented and the robot is able to walk at 0.11 m/s. Literature [35] considers the problem of joint trajectory control with oscillations cancelling for planar multilink flexible manipulators. A feed-forward torque is designed to preset the elastic coordinates of the system, so that the controller is able to drive the arm along the desired trajectory. In [36], in order to reduce gait planning and to get a good tracking performance, master-slaver dual-leg coordination control was proposed for the biped robot with heterogeneous legs. In [37], an open-closed-loop iterative learning control algorithm with angle correction term is proposed, which uses the error signal and the deviation of two adjacent error signals to adjust itself.

In this work, the proposed neurodynamics-based method is an open-loop controller. As shown in Figure 1, control method based on energy-efficiency optimization is proposed to obtain the trade-off between the force-moment equilibrium and energy efficiency for biped walking by minimizing the energy-related cost function while guaranteeing physical constraints. There are two parts in the control system: QP problem for energy optimization and a LVI-based primal-dual neural network for solving the QP problem. The control object of the proposed method is to follow the planned trajectories for the biped joints using optimal joint torques deduced by the proposed controller. The inputs of the controller include the planned trajectories for the biped joints, the force-moment equilibrium equation of biped robot, friction coefficient between the ground and the robot, and physical parameters of the motors equipped on the robot. The outputs of the controller are the optimal joint torques for the biped joints, which are deduced using a LVI-based primal-dual neural network.