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Abstract

Bioinspired quadruped robots are among the best robot designs for field missions over the complex terrain encountered in extraterrestrial landscapes and disaster scenarios caused by natural and human-made catastrophes, such as those caused by nuclear power plant accidents and radiological emergencies. For such applications, the performance characteristics of the robots should include high mobility, adaptability to the terrain, the ability to handle a large payload and good endurance. Nature can provide inspiration for quadruped designs that are well suited for traversing complex terrain. Horse legs are an example of a structure that has evolved to exhibit good performance characteristics. In this paper, a leg design exhibiting the key features of horse legs is briefly described. This leg is an underactuated mechanism because it has two actively driven degrees of freedom (DOFs) and one passively driven DOF. In this work, two control laws intended to be use in the stan ce phase are described: a control law that considers passive mechanism dynamics and a second law that neglects these dynamics. The performance of the two control laws is experimentally evaluated and compared. The results indicate that the first control law better achieves the control goal; however, the use of the second is not completely unjustified.

Keywords

Bioinspired Robots Impedance Control Agile Locomotion Underactuated Robots

1. Introduction

As society evolves, its technological requirements also evolve. In particular, modern societies require robots that can safely negotiate complex terrain for a variety of applications, such as extraterrestrial exploration or search and rescue operations in disaster scenarios caused by natural and human-made catastrophes, such as those caused by nuclear power plant accidents and radiological emergencies. In general, wheeled vehicles may not be able to negotiate these types of terrain (e.g., when there is a very complex terrain). Legged animals provide evidence that legs may be necessary for traversing complex terrain [1].

Several legged robots have been built [1, 2, 3, 4, 7, 5]; however, only a few are capable of achieving agile locomotion [8, 4]. One probable reason for this deficiency [9] is that the most common approach to building legged machines, from 19th-century wind-up toys to many of the modern legged robots, is to mimic the joint trajectories of animals.

However, this approach has become less common over the last decade, and a new generation of biologically inspired robots is emerging [1, 8, 10]. In addition to mimicking the joint angle trajectories of animals, these robots attempt to emulate the key features that distinguish the dynamic behaviour of animals. One of the main characteristics of these robots that allows them to mimic the springy tendons in animals is the use of springs and other passive elements to simplify the control of the leg and to conserve energy [10, 11]. Furthermore, control methods are being developed to take advantage of the dynamic characteristics of these systems [10, 11], where several researchers are building systems that do not require any level of control [9]. Most of these studies have been performed with the general goal of increasing energy efficiency and thus improving autonomy and endurance. However, these designs are v ery sensitive to disturbances.

In contrast, most modern robotic applications, influenced by traditional industrial robotics, use a completely different approach in which the natural dynamics of the mechanism are cancelled out to achieve the precise following of the desired trajectory.

Nature can provide inspiration for the design of legged robots that are suited for traversing complex terrain. One source of inspiration is agile quadrupeds, of which horses are among the most efficient. Horse legs have evolved to provide speed, endurance and strength superior to the legs of any other animal of equal size [12]. The adaptation of horse legs for agile performance has resulted in longer legs relative to body size than those of similar quadrupeds, resulting in a longer stride length. Horse legs are relatively lightweight and have a mass distribution that improves the leg's oscillation frequency [12]. The horse leg's kinematic structure optimizes the use of its joints for bearing loads. Endurance for locomotion results from the economy of effort achieved by the storage of elastic energy in the tendons during certain phases of the locomotion cycle. This energy is returned during the more demanding phases of the cycle.

Following these examples that nature has provided, one can anticipate the substantial advantages of controlling the motion of a bioinspired horse-like leg. However, controlling the motion of a mechanism that includes both active and passive elements, which must be synchronized and feature different inherent dynamics, is not straightforward.

Previous works on legged locomotion that combine active and passive actuation (i.e., using passive elements as actuators such as springs) can be classified as limit-cycle walkers [13, 14] and force-controlled underactuated legged robots. The former type focuses on the energy efficiency advantage of passive dynamics and uses active elements to inject the instantaneous dynamic requirements that are not provided by passive elements to maintain the robot in a limit-cycle. The latter type is more suited for complex, natural environments due to their robustness to unexpected perturbations, which is an advantage over limit-cycle walkers.

Force-based control of underactuated robots taking advantage of elastic elements has been addressed by Grizzle and his collaborators [15, 16, 17], who proposed the hybrid zero dynamics (HZD) approach, which has been shown to be robust to moderate terrain unevenness and stiffness [18]. However, rather than taking advantage of the system dynamics, the HZD approach converts the system (through virtual constraints) to a certain target model and then applies control laws that are suitable for the new model. Nevertheless, HZD has been adapted to preserve the dynamics of passive elements in the case of MABEL [15, 19], which is used to modify only a portion of the robot dynamic behaviour [15] and accommodates its behaviour to the passive element to achieve asymptotically stable behaviour.

Nonetheless, more intuitive control methods for using compliant actuation systems have been proposed [6, 20]. Whereas the hybrid zero dynamics approach relies on a detailed dynamic modelling of the robot, Pratt's virtual model control [21, 22] presents an approach that controls active elements so that they behave as passive elements. However, this method is not directly applicable to control an underactuated mechanism because it uses virtual passive elements on each joint and in task space to achieve its desired behaviour, although this work is based on Pratt's approach.

However, virtual model control requires an ideal behaviour of the active elements to reproduce the desired robot dynamics. This problem is augmented when a real passive element is inserted in series with active elements.

There are many reasons why the robotics community has studied underactuated robots. Previous studies have demonstrated that a sufficient condition for an underactuated robotic manipulator is that the number of passive joints is less than or equal to the number of active joints [23, 24], where the output controlled is the position or exerted force of the end effector (in this case the robot's foot). Control strategies have been proposed to control systems that exploit the systems' intrinsic dynamic coupling [23, 25, 26] in both the joint and task spaces. These approaches have been implemented on legged robots [27, 28]. Such strategies allow passive joints to move freely until the desired joint position is achieved; the passive joint is then blocked, allowing the other joints to move such that the overall desired position can be achieved. However, for the leg presented in this paper, such strategies cannot be directly applied because the passive joint cannot move freely due to its attached spring.

In this study, a force-based impedance control scheme [29] is implemented on a bioinspired underactuated leg prototype (fig. 1). This prototype is a planar 3-DOF leg directly driven by two series elastic actuators (SEAs), one at the hip and the other at the knee. The fetlock joint is passively driven by a spring resembling the superficial digital flexor in horses [30]. The impedance control law was motivated by the fact that animals appear to change the apparent stiffness of their legs to accommodate the terrain [31, 32]. The objective of impedance control is not to directly control position or force, but the relationship between them. This allows reducing or increasing apparent stiffness, damping or mass depending on the task. Moreover, with a single control scheme it is possible to control both position and force, although not directly. In addition, it is important to note that the control schemes studied in this work are intended to reduce impact forces when the legs hits the ground and to modify the apparent leg stiffness during the support phase.

Figure 1.

The HADE leg prototype: conceptual design (left) and prototype (right)

This paper experimentally compares the performance of two control approaches applied to an underactuated mechanism. The goal of this comparison is to determine the extent to which it is necessary or desirable to take advantage of passive element dynamics to achieve a certain control objective. This issue is important because underactuated mechanisms complicate the controller design significantly and because underactuated mechanisms cannot follow arbitrarily defined trajectories [33], thus creating a limit to the workspace of the leg. This can be a disadvantage in the case of a legged robot because it is desirable that the leg is capable of following any particular trajectory to avoid collisions or to reach a certain foothold.

Because field robots are intended to work in unstructured environments, interaction control schemes are used in this paper. In this study, two control laws will be applied to the leg and their performance will be compared. One control law incorporates the dynamic effects of the spring, and the other law neglects these effects. The objective of this investigation is to study the effect of the spring dynamics in the performance of the leg, in terms of a desired impedance.

In this paper, the design of the HADE leg [8], which exhibits the key features of a horse leg, is briefly described in Section 2. Section 3 presents the two control laws. In Section 4, the proposed control law is experimentally validated, and the performance of the two control laws is compared. Finally, Section 5 provides the conclusions and future work.

2. The HADE Leg Design

The design of the HADE leg has been described in detail in previous publications. In this paper, a brief summary of the design is included for the sake of completeness, and the interested reader is referred to [34, 8] for a complete discussion on the leg's mechanical design.

The HADE leg was designed to try to achieve the horse leg's superior speed, endurance and strength in an artificial prototype. Five key elements affect the performance of the horse leg:

Effective leg length. A longer effective length increases stride length, which improves leg speed and reduces the energetic cost of locomotion. The average effective leg length of an Arabian horse is 60% of the longitudinal horse length from nose to tail [35].

Leg kinematics. Movement in the equine limbs occurs predominantly in the sagittal plane; these kinematics are energetically advantageous in cursorial species [36].

Mass distribution. The leg's mass distribution affects its natural frequency of movement and thus the horse's speed. Horses bred for running have 80%-90% of their leg mass located near the hip; this percentage is lower in other breeds. This property increases the natural frequency of the leg motion and allows for a higher stride frequency [37].

Muscle power. The leg's muscle power capacity influences joint speed, torque capacity and limb strength.

Elasticity. Elastic energy storage in the tendons provides elastic recoil, reducing the muscles' power requirements during the more energetically demanding phases of movement and improving endurance [38].

The biomimetic design of the 3-DOF planar leg, shown in Figure 2, accounts for these five elements. The leg is actuated at the hip and knee by two series elastic actuators (SEAs), and the fetlock is passively driven by a spring. The actuators represent the muscles in a horse's hip and knee, and the fetlock spring emulates the superficial digital flexor.

Figure 2.

Biomimetic model of a leg for agile locomotion. The series elastic actuators are labelled as SEA, and SDF is the superficial digital flexor

SEAs were selected as the driving actuators for the leg's active joints because the SEA behaviour resembles the muscles' mechanical responses and the neuromuscular control system's behaviour [39].

The direct kinematics of the leg is described by the following equations for a 3-DOF planar manipulator (see Figure 1):

\begin{array}{l} x_{b} = - a_{1} s i n (q_{1}) - a_{2} s i n (q_{1} + q_{2}) - \\ a_{3} s i n (q_{1} + q_{2} + q_{3}) \end{array}

(1)

\begin{array}{l} z_{b} = - a_{1} c o s (q_{1}) - a_{2} c o s (q_{1} + q_{2}) - \\ a_{3} c o s (q_{1} + q_{2} + q_{3}) \end{array}

(2)

ϕ = q_{1} + q_{2} + q_{3}

(3)

where $a_{1}$ , $a_{2}$ and $a_{3}$ are the lengths of the thigh, crus and hoof links, respectively; x, y and ϕ are the Cartesian hoof position and orientation in the leg base reference frame [ $x_{o}$ , $y_{o}$ ] (see Figure 3); and $q_{i}$ , $i = 1,2,3$ are the hip, knee and fetlock joint coordinates, respectively, where the zero position is when the leg is straight down, i.e., when all of the links are parallel to the $X_{0}$ axis. From these equations, the Jacobian matrix can be derived as follows:

Figure 3.

Graphical representation of the control objective

\begin{array}{l} J = \\ [\begin{array}{r} - a_{1} C_{1} - a_{2} C_{12} - a_{3} C_{123} & - a_{2} C_{12} - a_{3} C_{123} & - a_{3} C_{123} \\ a_{1} S_{1} + a_{2} S_{12} + a_{3} S_{123} & + a_{2} S_{12} + a_{3} S_{123} & + a_{3} S_{123} \\ 1 & 1 & 1 \end{array}] \end{array}

(4)

where,

\begin{array}{l} C_{1} & = & c o s (q_{1}), \\ C_{12} & = & c o s (q_{1} + q_{2}), \\ C_{123} & = & c o s (q_{1} + q_{2} + q_{3}), \\ S_{1} & = & s i n (q_{1}), \\ S_{12} & = & s i n (q_{1} + q_{2}), \\ S_{123} & = & s i n (q_{1} + q_{2} + q_{3}) \end{array}

(5)

The dynamics of the HADE leg is derived here as follows: first of all let us recall the expression of the torque for a robotic manipulator with elastic joints [?]:

H (q) \overset{..}{q} + C (\dot{q}, q) \dot{q} + G (q) + K (q - θ) = 0

(6)

I_{m} \overset{..}{θ} + J^{T} F - K (q - θ) = τ

(7)

where $H (q)$ is the inertia matrix of the manipulator, $C (\dot{q}, q)$ is the centrifugal and Coriolis torques, $G (q)$ is the gravitational induced torque, J is the Jacobian matrix, F is the exerted force at the foot, $I_{m}$ is a diagonal matrix containing the rotor's inertia ( $I_{i}$ ) multiplied by the square of the gear reduction ratio ( $r_{i}$ ), i.e.,

I_{m} = [\begin{matrix} I_{1} r_{1}^{2} & 0 & 0 \\ 0 & I_{2} r_{2}^{2} & 0 \\ 0 & 0 & I_{3} r_{3}^{2} \end{matrix}]

(8)

and K is a diagonal matrix containing the spring constants of each joint, i.e.,

K = [\begin{matrix} k_{1} & 0 & 0 \\ 0 & k_{2} & 0 \\ 0 & 0 & k_{3} \end{matrix}]

(9)

where the expression $H (q) \overset{..}{q} + C (\dot{q}, q) \dot{q} + G (q)$ can be seen as the rigid part of the robot [?]. If we consider this to be the torque applied by the actuators just before the elastic joint $τ_{a}$ , and consider that $τ_{a} = τ_{m} r$ and $\dot{q} = {\dot{q}}_{m} / r$ , where $τ_{m}$ is the torque applied by the motor, ${\dot{q}}_{m}$ is the motor's angular velocity and r is the gear reduction ratio we have that

H {\overset{..}{q}}_{m} R^{2} + C (\dot{q}, q) {\dot{q}}_{m} R^{2} + G (q) R = τ_{m}

(10)

where

R = [\begin{matrix} \frac{1}{r_{1}} & 0 & 0 \\ 0 & \frac{1}{r_{2}} & 0 \\ 0 & 0 & \frac{1}{r_{3}} \end{matrix}]

where $r_{1}$ and $r_{2}$ are around 3000 and $r_{3}$ is 1. From equation 10 it is clear that the torques due to inertia and Coriolis of the first two links are six orders of magnitude smaller than the torques of the motors and that the gravitational terms are three orders of magnitude smaller. Thus, one need only be concerned about the dynamics of the third link (i.e., the passively driven joint).

3. Impedance Control Strategy

This study investigates the value of incorporating the effect of a spring-driven joint in the impedance control law for an underactuated leg. We compare two control approaches. The first approach considers the leg to be fully actuated, i.e., the control law only includes the two actuated degrees of freedom, and the torques and displacements produced by the spring are treated as disturbances in the control loop. The second approach accounts for all three of the legs' DOFs and includes a feed-forward term to the two active joints to account for the torque produced by the spring. The following control objective is used for both control laws:

F (s) = Z (s) X (s) = (K + B s) X (s)

(11)

where $F (s)$ is the force vector, $X (s)$ is the position vector in task space coordinates, $Z (s)$ is the mechanical impedance, and K and B are the stiffness and damping matrices, respectively, and s is the Laplace transform variable. Figure 3 shows a graphical representation of this objective. As shown in the figure, the control objective is to modulate the vertical and horizontal forces (in the leg-base coordinate frame) as a spring-damper system.

The following two control laws, each designed to achieve the control objective, are applied to the leg.

3.1. Force-based impedance control law neglecting the spring dynamics (NSD)

The first control law considers the leg to have two DOFs and treats the passive joint as a disturbance. The reference forces are given by

F_{r 2} = [\begin{matrix} F_{x r} \\ F_{z r} \end{matrix}] = [\begin{matrix} - K (x - x_{r}) - B (\dot{x} - {\dot{x}}_{r}) + F_{x d} \\ - K (z - z_{r}) - B (\dot{z} - {\dot{z}}_{r}) + F_{z d} \end{matrix}]

(12)

where $F_{r 2}$ is the reference force vector; $x_{r}$ and $y_{r}$ are the vertical and horizontal reference positions, respectively; $F_{x r}$ and $F_{x d}$ are the force component and desired force in the x direction; $F_{z r}$ and $F_{z d}$ are the force component and desired forces in the z direction; and K and B are the stiffness and damping constants, respectively, that define the desired mechanical impedance of the leg. It is worth noticing that $F_{x d}$ and $F_{z d}$ are the desired forces when the position error is zero; otherwise, the forces can be seen as a bias term for the force, which is accommodated by the stiffness and damping forces in the control scheme.

The reference torques are given by the following equation, which maps the task space forces to the joint torques:

τ = J_{2}^{T} F_{r 2}

(13)

where the subscripts indicate that $F \in R^{2}$ and that $J_{2}$ is the 2 × 2 Jacobian matrix defined as follows:

J_{2} = [\begin{matrix} - a_{1} C_{1} - a_{2} C_{12} & - a_{2} C_{12} \\ a_{1} S_{1} + a_{2} S_{12} & a_{2} S_{12} \end{matrix}]

(14)

Equations (12) and (13) lead to the control scheme shown in Figure 4.

Figure 4.

Force-based impedance control scheme in which the spring dynamics are neglected

It is important to note that the NSD controller assumes that the leg's tip is the ankle. The importance of this assumption will be discussed in detail below.

3.2. Force-based impedance control law incorporating the spring dynamics

The second control law considers the leg to have three DOFs. The control law assumes that the passively driven joint is uncontrolled; however, a feed-forward term is included in the reference forces for the active joints. Consider that:

τ_{r} J^{T} F = [\begin{matrix} J_{1,1} F_{x r} + J_{2,1} F_{z r} + J_{3,1} τ_{ϕ} \\ J_{1,2} F_{x r} + J_{2,2} F_{z r} + J_{3,2} τ_{ϕ} \\ J_{1,3} F_{x} + J_{2,3} F_{z} + J_{3,3} τ_{ϕ} \end{matrix}]

(15)

where $τ_{ϕ r}$ is the reference torque about an axis perpendicular to the leg and that is located at the tip of the foot. Notice that the Jacobi transposed mapping requires the third joint (i.e., the ankle joints) to exert a torque even if we set $τ_{ϕ r} = 0$ . As mentioned earlier, this variable is not possible to control as the last joint is passively driven by a spring, thus the objective is to make the controller output equal to zero at all times while taking in to consideration the spring torque with the other two actuators. To do this, we choose to make $τ_{3 r} = τ_{3}$ and solve for $τ_{ϕ r}$ which yields:

τ ϕ r = (τ_{3} - J_{1,2} F_{x r} - J_{2,3} F_{z r}) \frac{1}{J_{33}}

(16)

thus, the reference forces are given by:

F_{r 3} = [\begin{matrix} F_{x r} \\ F_{z r} \\ τ_{ϕ r} \end{matrix}] = [\begin{matrix} - K (x - x_{r}) - B (\dot{x} - {\dot{x}}_{r}) + F_{x d} \\ - K (z - z_{r}) - B (\dot{z} - {\dot{z}}_{r}) + F_{z d} \\ (τ_{3} - F_{x r} J_{1,3} - F_{z r} J_{2,3}) \frac{1}{J_{3,3}} \end{matrix}]

(17)

where $τ_{ϕ r}$ is the desired torque for the hoof orientation; $τ_{3}$ is the torque exerted at the fetlock joint; and $J_{i j}$ are the components of the Jacobian matrix. The remaining terms are the same as those in equation (12). The reference torque $τ_{ϕ r}$ is chosen in this form because when mapping the reference forces to the joint space, the term vanishes from the equations, leaving only the measured torque $τ_{3}$ as a reference in the third joint. The reference torques are also given by the equation that maps the task space forces to the joint torques as follows:

τ_{r} = J^{T} F_{r 3}

(18)

where the subscripts indicate that $F_{r 3} \in R^{3}$ and that $J_{3}$ is a 3 × 3 matrix, which is the same as in equation (4). When the terms on the right side of equation (17) and in $J_{i j}$ are replaced with their corresponding values from equation(4) in this equation, after some manipulation, the reference torques ( $τ_{r i}$ ) can be written as:

τ_{r} = [\begin{matrix} τ_{r 1} \\ τ_{r 2} \\ τ_{r 3} \end{matrix}] = [\begin{matrix} F_{x r} {\hat{J}}_{1,1} + F_{z r} {\hat{J}}_{2,1} - τ_{3} \\ F_{x r} {\hat{J}}_{1,2} + F_{z r} {\hat{J}}_{2,2} - τ_{3} \\ τ_{3} \end{matrix}]

(19)

where,

\begin{array}{l} {\hat{J}}_{11} & = & - a_{1} S_{1} - a_{2} S_{12} \\ {\hat{J}}_{12} & = & - a_{2} S_{12} \\ {\hat{J}}_{21} & = & a_{1} C_{1} + a_{2} C_{12} \\ {\hat{J}}_{22} & = & a_{2} C_{12} \end{array}

(20)

As can be observed in the above equations, the reference given to the third joint is the exerted torque $τ_{3}$ ; thus, this joint is allowed to move freely and its contribution is reflected in the active joints.

The control scheme described in this section, based on equations (17) and (19), is depicted in Figure 5.

Figure 5.

Force-based impedance control scheme in which the spring dynamics are included

3.2.1. Applicability of the impedance control law incorporating the spring dynamics

Several important remarks should be made concerning this control law. First, assuming that the PID controller accurately tracks the reference torques, mapping the achieved torques to the task space forces yields

\begin{array}{l} {\hat{F}}_{x} = τ_{3} (c o s (2 q_{1} + q_{2}) + s i n (2 q_{1} + q_{2}) + \\ c o s (q_{2}) + s i n (q_{2}) - a_{3} s i n (q_{1} + q_{3}) + \\ a_{3} s i n (q_{1} - q_{3})) \frac{1}{2 a_{1} c o s (q_{1}) s i n (q_{2})} \end{array}

(21)

\begin{array}{l} {\hat{F}}_{z} = τ_{3} \frac{c o s (q_{1})}{a_{2} s i n (q_{2})} + \\ \frac{a_{2} c o s (q_{1} + q_{2}) + a_{1} s i n (q_{1}) - a_{1} a_{3} s i n (q_{2} + q_{3})}{a_{1} a_{2} s i n (q_{2})} + \\ (a_{2} s i n (q_{2}) + a_{2} s i n (2 q_{1} + q_{2}) + a_{2} a_{3} s i n (q_{1} - q_{3}) - \\ a_{2} a_{3} s i n (q_{1} + q_{3})) \frac{1}{2 a_{1} a_{2} c o s (q_{1}) s i n (q_{2})}) \end{array}

(22)

where ${\hat{F}}_{x} = F_{x} - F_{x r}$ and ${\hat{F}}_{z} = F_{x} - F_{z r}$ . These equations imply that there will be a steady-state error dependent on both $τ_{3}$ and the leg configuration. There are two ways to eliminate this error:

Setting $τ_{3} = 0$ which is the general case for a leg in the swing phase and

Selecting a leg configuration that causes the terms other than $τ_{3}$ to become zero.

We will discuss the second alternative, which requires the following set of equations to be solved:

\begin{array}{l} 0 = (c o s (2 q_{1} + q_{2}) + s i n (2 q_{1} + q_{2}) + c o s (q_{2}) + s i n (q_{2}) - \\ a_{3} s i n (q_{1} + q_{3}) + \\ a_{3} s i n (q_{1} - q_{3})) \frac{1}{2 a_{1} c o s (q_{1}) s i n (q_{2})} \end{array}

(23)

\begin{array}{l} 0 = \frac{c o s (q_{1})}{a_{2} s i n (q_{2})} + \\ \frac{a_{2} c o s (q_{1} + q_{2}) + a_{1} s i n (q_{1}) - a_{1} a_{3} s i n (q_{2} + q_{3})}{a_{1} a_{2} s i n (q_{2})} + \\ (a_{2} s i n (q_{2}) + a_{2} s i n (2 q_{1} + q_{2}) + a_{2} a_{3} s i n (q_{1} - q_{3}) - \\ a_{2} a_{3} s i n (q_{1} + q_{3}) \frac{1}{2 a_{1} a_{2} c o s (q_{1}) s i n (q_{2})} \end{array}

(24)

Solving these equations defines a workspace with perfect force tracking. Figure 6 shows this workspace, which is computed by assuming that $q_{3} = - 1.8$ and numerically solving equations (23) and (24). As shown in the figure, this workspace (represented by the shaded area) includes not only the leg configurations in which we are interested but also other configurations that are not of interest in this application. However, the other configurations may be of interest in other applications, such as manipulation.

Figure 6.

Workspace of the control law for $q_{3} = - 1.8$ rad.

Additionally, if a wider workspace is needed because the form of the steady-state error is known, the control law could be modified to include a feed-forward term to pre-compensate for this error. This approach is commonly used in the control of flexible manipulators [40].

The second important remark is that this control law is based on the implicit assumption that the flexible behaviour (in the directions of motion) of the spring during free motion can be neglected. This assumption is valid for springs that are sufficiently rigid to maintain $q_{3} = 0$ under free-motion conditions. This assumption also imposes a practical limitation on the use of this control law. If this condition is not met, the flexible behaviour of the third joint must be considered and such approaches as those described in references [40, 41, 42] should be used. The problem may also be interpreted using the analogy between a spring and capacitor (as both a spring and a capacitor can be described using the same differential equations). A spring acts as a mechanical low-pass filter with a cutoff frequency that depends on the inverse of the stiffness. Consequently, the mechanical behaviour of the spring will be more similar to a rigid body with higher s pring stiffness.

3.3. Discussion

Note that in the above analysis, the role of the SEAs springs was neglected because the spring constant of the actuators is 400 times larger than the higher spring constant of the spring attached to the fetlock joint. Consequently, the dynamic effects of the SEAs springs are much faster than those of the fetlock spring and can thus be safely neglected.

In addition, note that the actuators are force controlled; thus, the joint position error is not considered by the internal PID controller. However, both control laws account for the position error in task space (see equations 12 and 17), and this error is transformed into a force error, which is still corrected by the control law, as with any force-based impedance control [29].

Regarding the definition of the force and position references, this controller intends to be a lower level control for a walking robot, where the position and force references are generated in a higher level and then passed to the controller. The desired forces ( $F_{x d}$ and $F_{z d}$ ) should be chosen according to the locomotion phase; for example, during the stance phase, the vertical forces should be chosen considering the force distribution among the legs to support the robot and its payload weight, whereas the horizontal forces must be chosen to be inside the friction cones to prevent slippage. In contrast, during the swing phase, the force reference should be set to zero to prevent damage on the robot or the environment if the robot crashes with an obstacle.

4. Experiments and Results

To test the proposed control method incorporating the spring dynamics, the control scheme (Figure 5) was implemented in the HADE leg prototype (Figure 1).

4.1. Experimental setup

Figure 7 shows the experimental setup used in this study. The leg (see Figure 1) is fixed at the hip to a rail that allows for vertical movement of the base (hip) and restricts horizontal movements. The rail is fixed to a wall and has a mechanical limit to prevent falling. Moreover, the foot is placed on the surface of a motorized treadmill to allow for horizontal displacement of the foot against the surface.

Figure 7.

Experimental setup

Each leg actuator included encoders to measure joint position and force, and the fetlock's spring deflection was measured by a linear encoder. The control law and data acquisition were implemented with a National Instruments NI PXI-1042Q with a control loop execution time of 500 μ s.

The exerted force at the foot was computed using the measured force on each joint using the formula $F = J^{- T} τ$ , where F is the vector of task space forces, J is the Jacobian matrix of the leg (see (4)) and τ is the vector of the torques in joint space. It is important to note that to obtain a meaningful comparison, the equations used to compute the exerted force and position of the foot are the same.

4.2. Influence of the leg dynamics

Before validating the proposed control laws it is important to remark that neither of the control schemes consider the complete leg dynamics. This simplification can be done because we have corroborated both analytically (see Section 2) and experimentally that the influence of the leg dynamics is not significant in the first two joints. The experimental evaluation is the subject of this subsection.

To evaluate the influence of the leg dynamics we have conducted a series of experiments with the leg suspended (in order to eliminate the $J^{T} F$ term). To do the experiments, a position control law was implemented to the leg (see Figure 8). Moreover, each experiment consists of 10 repetitions and the error values here indicated are the average value of all the performed tests.

Figure 8.

Position controller to evaluate leg dynamics

First, to evaluate the gravitational component the leg is commanded to follow a variety of trajectories covering the whole workspace, ensuring low, constant speed and no acceleration. The observed variation of the controller error among trajectories was 3 % for the hip joint and 2 % for the knee joint. These errors are not significant enough to consider the system as dynamical.

Second, to analyse the effect of the Coriolis term ( $C (q_{,} \dot{q})$ ), a trajectory covering the whole workspace of the leg was commanded to the leg at two constant and opposite velocities (low: 0.5 m/s and high: 1.5 m/s), and the error in steady state was analysed during the time the joint velocity was constant to avoid exciting inertial terms. In this experiment, the variation of the error was 0.5 % for the knee and 1 % for the hip. Once more, these errors are not meaningful enough to consider a controller that includes all the leg dynamics.

Finally, to evaluate the influence of the inertial terms, the same trajectory used for the experiment of the Coriolis term was used, this time using a constant acceleration of value 0.5 m/ $s^{2}$ and 2 m/ $s^{2}$ for each of the two trajectories. The observed variation of the error between the two experiments was 2 % for the hip and 1 % for the knee, which is not as enough as to consider a controller that considers the full leg dynamics. Table I shows the variation of the error for the joints for each of the components evaluated in equation (18). As it can be seen in the table, the variation between experiments is very small, thus the dynamics of the leg does not introduce significant dynamical effect in its behaviour. This fact can be explained because the joint transmission system features a very high reduction ratio (around 3000) for the first two joints, which makes the actuator dynamics dominate the system behaviour. Nevertheless, when the leg is in contact with the ground, the dyn amic contribution of the third joint remains an issue. Therefore, in this study we apply two control laws, one including this dynamic and the other neglecting it in order to investigate the effects of the aforementioned dynamics in the control objective.

Table 1.

Error variation of the leg

Error variation	Hip	Knee	Task Space
$H (\overset{..}{q})$	3 %	2 %	3.57 %
$C (\dot{q}, q)$	1 %	0.5 %	1.05 %
$G (q)$	2 %	1%	3 %

4.3. Control law validation

In this section, we will experimentally validate the applicability of the ISD control scheme by testing its performance on the leg using two different SDF spring stiffness coefficients to demonstrate that the control law can accommodate variations in the spring stiffness. Note that the primary goal of the control law is to control the vertical leg stiffness, thus a certain error is expected in position and force both vertical and horizontal, though the vertical force error should perform better (within the applicable workspace) than the horizontal force. In the experiments, the leg was placed in an arbitrary initial position, with the foot in contact with the ground. The leg was commanded with a desired foot force pattern intended to illustrate the assumptions made in Section 3. A first spring with $K_{s 1} = 3,113$ N/m was used. The spring was then replaced by a spring with a spring constant of $K_{s 2} = 6,670$ N/m; the experiment was repeated with the control structure unaltered and the leg in the same initial position. The desired force pattern was then used again as the command for the leg.

Figure 9 shows the pattern of the commanded vertical foot forces for the leg (dashed line), the measured force when using $K_{s 1}$ (thin solid line), and $K_{s 2}$ (thick solid line). The first command to the leg was a downward force pattern beginning from $F_{z} = 0$ N, and then, applying a series of −10 N step signals every 4 s until $F_{z} = - 50$ N at $t = 24 s$ . The results demonstrate that the proposed control law is capable of controlling the leg behaviour with any spring stiffness if the assumptions made in Section 3 are met. This is evidenced when the leg reaches the value of −50 N when it is pushing downwards. Here, the leg moves as it reaches the threshold when the leg is capable of lifting its base (the hip) from the mechanical top that prevents falling. Next, when the leg starts exerting horizontal forces it reaches again the applicable workspace, thus reaching the target force again ( $t = 23 s$ ).

Figure 9.

Vertical force reference (dashed line) and vertical force with an ISD controller using $K_{s 1} = 3,113$ N/m (thin solid line) and $K_{s 2} = 6,670$ N/m (thick solid line) vs. time

Similarly, Figure 10 shows the pattern of the commanded horizontal foot forces for the leg (dashed line) and the controller response using $K_{s 1}$ (thin solid line) and $K_{s 2}$ (thick solid line). The leg was commanded to maintain a horizontal force equal to zero until $t = 24$ s (while the vertical force pattern was commanded), after which the same pattern described for the vertical force was commanded to the leg. In the figure we can observe a drift in the commanded force, this is because the leg is resting over a treadmill with a very high friction coefficient, which the leg is not capable of moving by itself at the commanded force values. Nevertheless, it still follows the force relationship that was commanded (keep in mind that we are controlling the relationship between force and position and not position or force separately), in other words, the force pattern described by the leg has the same shape as the commanded force though it cannot follow the reference exactly.

Figure 10.

Horizontal force reference (dashed line) and vertical force with an ISD controller using $K_{s 1} = 3,113$ N/m (thin solid line) and $K_{s 2} = 6,670$ N/m (thick solid line) vs. time

It can be seen from the figures that the tracking performance of the controller is almost the same for both springs used. The results show that the disturbances created by moving the leg in one coordinate direction produce a chattering behaviour in the other direction (i.e., $t \approx 34 s$ ). In addition, because there was no compensation for gravitational effects, the vertical component was more sensitive to these disturbances. Nevertheless, when using a stiffer spring, the chattering behaviour was significantly reduced.

These results show that the control law accomplishes the objective of tracking the reference force despite the changes in the spring's stiffness. When the stiffer spring was used, the impact peaks that occurred when the leg came in contact with the ground were reduced. Thus, the stiffer spring was used in the experiments to compare the two proposed control laws.

As a final comment on the experiments, note that the vertical force should be negative to maintain contact, which means that a positive force would ultimately lead to separation; however, because the foot force was not measured by a force sensor but calculated from the joint torques with the relationship $F = J^{- T} τ$ , the contact events were detected by the fetlock spring deflection. In the above experiments, the less stiff spring was also disadvantageous because it oscillated for a longer time than the stiffer one, thus making it more difficult to detect contact and separation. Moreover, an isolated positive peak, such as the peak present in8 when $t = 33$ s, in the exerted force at the foot does not necessarily indicate separation.

4.4. Control comparison

The objective of this section is to determine the extent to which it is necessary or desirable to take advantage of the spring dynamics to achieve a certain control objective. Here, we compare the experimental performance of the control laws described above. To accomplish this objective, two experiments were performed. In the first experiment, the leg was placed in the same initial position used for the previous experiments and a sweeping force pattern, consisting of a series of step signals from 0 N to −100 N of 1 s each, was commanded in each direction ( $F_{x}$ and $F_{z}$ ) separately. This experiment was performed with the control scheme shown in Figure 3, in which the spring dynamics were neglected using values of $K = 0,100,…,900$ . The damping constant B equals 0 when $K = 0$ and 0.1 for all the other values of K. The controller structure was then changed to the second controller shown in Figure 5 (using the same PID constants for comparison), and the procedure was rep eated.

To produce a meaningful comparison, when computing the vertical and horizontal forces and position values for the NSD control law, the full 3-DOF kinematic model of the leg and the measured values of $τ_{3}$ and $q_{3}$ were considered. Moreover, the experiments were repeated 10 times for each value of K and for each controller.

Figures 11 and 12 show the average mean square error for the leg's vertical and horizontal positions, respectively, where the results obtained with the ISD and NSD controllers are shown as a solid line and dashed line, respectively. In the figures, it can be seen that as the value of K increases, the performance of each controller is fairly similar. However, for low values of K, the NSD controller presents a lower position error (with the exception of $K = 0$ ) because the compensating term in the ISD controller does not cancel out the force bias term in the controller, thus preventing the leg from rearranging according to the fetlock spring torque. The case when $K = 0$ represents a pure force control, which allows the leg to move freely from the initial position to exert the desired force. In this case, the position error is not relevant; however, as shown in figures 11 and 12, the position error of th e NSD controller is greater because the fetlock joint torque was not compensated and the bias force term was not cancelled out, which exerted a torque that was higher than necessary to achieve the desired force.

Figure 11.

Vertical position error vs. K for the NSD (dashed line) and the ISD (solid line) control schemes

Figure 12.

Horizontal position error vs. K for the NSD (dashed line) and the ISD (solid line) control schemes

Figures 13 and 14 show the average mean square error for the leg's vertical and horizontal forces, respectively, where the results obtained with the ISD and NSD controllers are shown as a solid line and dashed line, respectively. It is important to note that the force error shown in the figures refers to the desired foot force $F_{d}$ , not to the joint force error. In the figures, the horizontal force (see Figure 14) presents a very similar behaviour for each controller for all values of K, whereas the vertical component shows a zigzag pattern for the NSD controller and a monotonically increasing pattern for the ISD controller. The difference between both behaviours is because the vertical force was more influenced by the fetlock spring; this behaviour was not considered by the NSD controller. In other words, the ISD controller more closely follows the control objective (as the stiffness increases, the force error shou ld be higher), where the spring dynamics has more influence.

Figure 13.

Vertical force error vs. K for the NSD (dashed line) and the ISD (solid line) control schemes

Figure 14.

Horizontal force error vs. K for the NSD (dashed line) and the ISD (solid line) control schemes

Nevertheless, the NSD controller exhibited a lower force error than the ISD controller because the PID loop rejected the spring contributions, whereas the ISD controller does not. This result indicates that in terms of the control objective (describing a desired impedance), the ISD controller outperformed the NSD controller.

It is worth noting that the leg's dynamics (gravitational and centrifugal terms) are not considered in the analysis, which makes the results dependent on the configuration. For that reason, the second experiment, performed to evaluate the controllers' performance, consisted of commanding a desired trajectory to the leg, which covers the anticipated workspace. Note that this is not the primary objective of the control law, and that the scheme is not intended to be used to position the leg in a desired location, but describe a force-position relationship when the leg is in contact with the ground. Thus the following experiments were only performed for comparison purposes. Note that the ground is located at $Z_{b} = - 1.07$ . Thus, the vertical error shown in the experiments when the leg is in the $Z_{b} \geq - 1.07$ position are not relevant to the intended use of the leg.

A square trajectory was selected as a reference because it involves decoupling the vertical and horizontal movements on the leg, which can be a serious problem when using manipulators with elastic elements [42]. Figure 15 shows the reference trajectory (dashed line) and foot trajectory with the ISD controller (thick solid line) and NSD controller (thin solid line) at a speed of 1 km/h. As shown in the figure, the behaviour of the two control approaches is very similar at low velocities. Figure 16 shows the reference trajectory (dashed line) and foot trajectory with the ISD controller (thick solid line) and NSD controller (thin solid line) at a speed of 2 km/h. In this figure, it can be seen that the trajectory-following performances of the controllers differ significantly. It is important to note that the leg makes contact with the treadmill when the foot coordinate is $Z_{b} = - 1.07$ , and whenever the foot vertical coordinate is lower than this value, the base is moving upward. This situation is not relevant because when the coordinate decreases, the leg is sustained by the mechanical top of the rail (Figure 7) and its relative position to the ground once more becomes $Z_{b} = - 1.07$ . At this point, one can observe how the ISD controller allows for more deviation from the trajectory than the NSD controller because the impact forces cancel out a portion of the reference joint torques (see equation (17)). This behaviour appears undesirable; however, it allows for a faster recovery after the impact and better tracking of the following horizontal phase of the trajectory.

Figure 15.

Reference trajectory (dashed line) and foot trajectory with the ISD controller (thick solid line) and NSD controller (thin solid line) at a speed of 1 km/h

Figure 16.

Reference trajectory (dashed line) and foot trajectory with the ISD controller (thick solid line) and NSD controller (thin solid line) at a speed of 2 km/h

As a final comment, it is important to note that at low velocities, both controllers exhibit the same performance; however, at higher velocities, which are when impact forces are important, compensating for the spring-produced torque allows the trajectory to be more closely followed. In other words, a controller that considers the elastic element on the leg performs better than one that neglects it.

5. Conclusions

This study presented an impedance control approach for a biomimetic robotic leg for agile locomotion. The leg's design is inspired by horse legs and exhibits the most important features, in the authors' opinion, of this animal model. The leg is driven by two SEAs, one at the hip joint and one at the knee joint. The fetlock is passively driven by a spring resembling the superficial digital flexor.

To control the leg motion and ground interaction, a control law was proposed based on the literature on robotic manipulators with passive joints. Two controllers were presented and described, one that neglects the spring dynamics and one that considers the spring dynamics. The workspace in which the steady-state error tends to zero was presented, and a means of extending the workspace to the full range of the leg workspace was discussed. Both proposed control laws were implemented in a real 3-DOF planar leg prototype, and experiments comparing their performance were conducted.

The results showed that both control laws achieve the expected performance in position error for an impedance control law at slow velocities. However, as the velocity increases, the control law neglecting the spring dynamics starts to fail because it does not account for the spring induced torques.

In summary, the control law incorporating spring dynamics outperforms the NSD control law. Nevertheless, the position tracking performance of the two control laws is very similar, for the majority of the impedances studied, the NSD controller exhibited a lower error in both position and force. Thus, even though the NSD controller was outperformed in terms of the impedance control objective, this controller is better suited when following position and force references alone is desired.

Footnotes

6. Acknowledgements

This work has been supported by Madrid Community through the project TECHNOFUSION (S2009/ENE-1679). Mr Juan Carlos Arevalo and Mr Manuel Cestari would like to thank the Spanish National Research Council and the Spanish Ministry of Economy and Competitiveness for funding their PhD research.

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On the Necessity of Including Joint Passive Dynamics in the Impedance Control of Robotic Legs

Abstract

Keywords

1. Introduction

2. The HADE Leg Design

3. Impedance Control Strategy

3.1. Force-based impedance control law neglecting the spring dynamics (NSD)

3.2. Force-based impedance control law incorporating the spring dynamics

3.2.1. Applicability of the impedance control law incorporating the spring dynamics

3.3. Discussion

4. Experiments and Results

4.1. Experimental setup

4.2. Influence of the leg dynamics

4.3. Control law validation

4.4. Control comparison

5. Conclusions

Footnotes

6. Acknowledgements

References