2. Previous approaches in bipedal robot control

In this section, we show a summary of some previous work published in journal publications regarding bipedal robot walking. This is shown in Table 1. The journal papers range from the years 2001 to 2019. The first column in the table lists the authors, while the title column shows the first three words of the paper title. Journal names are abbreviated according to listed standards from publishers. The table presents previous approaches in bipedal walking in terms of (a) frame of reference of the Jacobian used, (b) the number of joints in each leg, and (c) the method used in controlling bipedal motion. These are critical information toward evaluating the proposed modular relative Jacobian approach. Most of the work in bipedal walking used the body frame or world frame as the reference frame of the Jacobian. This influences the kind of control approach used in bipedal walking. We discuss below some of the more prominent methods of controlling bipedal walking below.

Method 1. Center of Mass (CoM) approach normally is paired with the Zero Moment Point (ZMP) such that the Jacobian is expressed w.r.t. the CoM point on the body, and the ZMP is computed from the torques generated based on the motion of CoM w.r.t. the ground.

Method 2. In the case of the whole-body control (WBC), different tasks, postures, and constraints, with their corresponding Jacobians, have different hierarchies of execution. All torques are added together at the joint-level control. In this case, the different Jacobians are expressed at their corresponding body frames. Each of the legs has 6-DOFs.

Method 3. The inverted pendulum approach uses the foot (base) frame as the reference frame for the Jacobian matrix. The center of gravity position and velocity is controlled to balance against gravity, as well as to ensure necessary torques are delivered on time to counter the gravitational force that is pulling the body downward. This approach is normally implemented on a 1-DOF leg platform.

Method 4. Ott–Grebogi–Yorke (OGY) is used to stabilize gait bifurcation. Jacobian is expressed at CoM. Pioncaré map is used to transform the continuous walking process, this is also known as the stride function. This approach normally studies 1-DOF leg platforms.

Method 5. The method of zero dynamics usually uses Jacobian linearization about a fixed point and deals with the invariance of results in the presence of disturbances. The reference frame of the Jacobian can be in the body or world frame. The legs can have 2-, 3-, or 4-DOFs.

Method 6. The operational space formulation in the bipedal robot is similar to the WBC. The reference frame of the Jacobian is in the body and each leg has 6-DOFs.

Method 7. Often used with gait bifurcation, gait stability is monitored through a threshold of the eigenvalues of the Jacobian. The frame of reference is the world frame and is usually implemented on a 1-DOF leg.

Overall, the major challenges on bipedal walking include gait stability, holistic control to address the issue of the hierarchy of motion execution, and the legs DOFs that will allow bipedal motion in full space. This paper does not address the issue of gait stability, but only addresses the issue of holistic control, the hierarchy of motion execution, and the legs DOFs in bipedal motion.

From Table 1, our proposed approach of relative reference frames is the only method that accommodates null-space control when each leg has 6-DOFs and the foot movement is controlled in the full space. This is also true when the foot movement is controlled in full-position control and each leg has 3-DOFs. Other recent studies in bipedal walking include stability constraints [Reference Sanchez, Wan, Kanehiro and Harada11], RRT-based motion planner [Reference Wasielica and Belter12], impact dynamics [Reference Luo, Xia and Zhu13,Reference Mobinipour14], and tactile feedback [Reference Guadarrama Olvera, Dean Leon, Bergner and Cheng15].

3. The modular relative jacobian for bipedal robots

The modular relative Jacobian formulation [Reference Jamisola and Roberts8] is shown here for bipedal robot implementation. Figure 1 shows the reference frames assignment in a bipedal robot. The labeled reference frames are the hip and foot frames of the bipedal robot. These correspond to the base and end effector frames, respectively, of the absolute Jacobian for each leg. Note that all joints are rotating about the z-axis (blue arrow) except those attached to the fixed reference frames $\{2\}$ and $\{4\}$ . For leg A, its absolute Jacobian is $\mathbf{J}_A={}^1\mathbf{J}_2$ . This means that the base frame for this Jacobian is frame {1} and its end effector frame is frame {2}. On the other hand, the absolute Jacobian for leg B is $\mathbf{J}_B = {}^3\mathbf{J}_4$ , with the base frame {3} and end effector frame {4}. Relative motion is expressed in terms of the motion of frame {4} w.r.t. frame {2}. Or in conventional terms, frame {2} as the base frame and frame {4} as the end effector frame. Thus the relative position and orientation can be expressed as, ${}^2\mathbf{x}_4=[{}^2\mathbf{p}_4,{}^2\boldsymbol{\Phi}_4]$ , where ${}^2\mathbf{p}_4$ is relative position and ${}^2\boldsymbol{\Phi}_4$ is relative orientation.

Using the above convention, the modular relative Jacobian, $\mathbf{J}_R$ , can be expressed in terms of the absolute Jacobians for each leg, as shown in [Reference Jamisola and Mastalli54], that is,

(1) \begin{equation}\mathbf{J}_R =\left[\begin{matrix}-{}^2\Psi_4 {}^2\Omega_1 {}^1\mathbf{J}_2&\quad \quad & {}^2\Omega_3 {}^3\mathbf{J}_4\end{matrix}\right]\end{equation}

where

(2) \begin{equation}{}^2\Psi_4 =\left[\begin{matrix}\mathbf{I} & -S({}^2\mathbf{p}_4) \\\mathbf{0} & \mathbf{I} \end{matrix}\right] \;\;\mbox{and}\;\;{}^2\Omega_1 =\left[\begin{matrix}{}^2\mathbf{R}_1 & \mathbf{0} \\\mathbf{0} & {}^2\mathbf{R}_1 \end{matrix}\right].\end{equation}

In general, $\mathbf{J}_A \in \mathbb{R}^{6\times n_A}$ and $\mathbf{J}_B \in \mathbb{R}^{6\times n_B}$ , where $n_A$ is the number of joints for leg A and $n_B$ is the number of joints for leg B; ${}^2\Psi_4, {}^2\Omega_1, {}^2\Omega_3 \in \mathbb{R}^{6\times 6}$ ; and $\mathbf{J}_R \in \mathbb{R}^{6 \times ({n_A+n_B})}$ . The symbol ${}^2\boldsymbol{\Omega}_3= f({}^2\mathbf{R}_3)$ is expressed as in (2). The relative position vector ${}^2\mathbf{p}_4$ is expressed as

(3) \begin{equation}{}^2\mathbf{p}_4 = {}^2\mathbf{p}_1 + {}^2\mathbf{R}_1 {}^1\mathbf{p}_3 + {}^2\mathbf{R}_3 {}^3\mathbf{p}_4.\end{equation}

To further simplify the expressions we introduce

(4) \begin{equation}\begin{matrix}\mathbf{Q}_A = -{}^2\Psi_4 {}^2\Omega_1 & \mbox{and} &\mathbf{Q}_B = {}^2\Omega_3\end{matrix}\end{equation}

such that the modular relative Jacobian becomes

(5) \begin{equation}\mathbf{J}_R =\left[\begin{matrix}\mathbf{Q}_A \mathbf{J}_A & \mathbf{Q}_B \mathbf{J}_B\end{matrix}\right].\end{equation}

In this implementation, we only intend to verify the accuracy of the kinematics model of the relative Jacobian for bipedal robots. Thus, we do not include the full dynamics information but added only the gravity term to the pure kinematics controller. This strategy will emphasize the results of our proposed solution on reduction of legs DOFs, holistic control, strict task prioritization, and modularity of expression. The total torques supplied to the joints are

(6) \begin{equation}\boldsymbol{\tau}_T =\mathbf{J}_R^T {\ddot{\textbf{x}}}_R +(\mathbf{I} - \mathbf{J}_R^T \mathbf{J}_R^{+T}) \boldsymbol{\tau}_O +\mathbf{g}_T.\end{equation}

The first term is the end effector motion, the second term is the null-space motion, and the last term is the gravitational term. Furthermore, $\boldsymbol{\tau}_T =\left[\begin{matrix}\tau_A^T & \tau_B^T\end{matrix}\right]^T$ where $\tau_A^T\in R^{n_A}$ and $\tau_B^T\in R^{n_B}$ , $\mathbf{g}_T =\left[\begin{matrix}\mathbf{g}_A^T & \mathbf{g}_B^T\end{matrix}\right]^T$ where $\mathbf{g}_A^T \in R^{n_A}$ and $\mathbf{g}_B^T \in R^{n_B}$ , ${\ddot{\textbf{x}}}_R\in R^{6}$ , and $\boldsymbol{\tau}_O \in R^{n_A+n_B}$ . Subscript A corresponds to leg A and subscript B corresponds to leg B and subscript T means total, that is, corresponding to both A and B. The symbol $\mathbf{g}$ is the gravity term, ${\ddot{\textbf{x}}}_R$ is the relative acceleration, $\boldsymbol{\tau}_O$ is the null-space control, and n is the leg DOFs.

4. Implementation of bipedal walking

It should be emphasized that in this implementation, there are two major controllers to perform bipedal walking: (1) the controller for the relative feet motion and (2) the controller to balance against gravity. The first controller corresponds to the world space control, while the second controller corresponds to the null-space posture control. To verify the proposed approach, we simulate a bipedal robot in Gazebo with ROS. Kinematics and dynamics information was derived using KDL libraries in ROS. The absolute Jacobians $\mathbf{J}_A$ and $\mathbf{J}_B$ for the legs were read directly from KDL, together with the information for the gravitational terms $\mathbf{g}_A$ and $\mathbf{g}_B$ . Because the implementation shown here is purely kinematics (with gravity compensation), its results are expected to be not as robust as when the dynamics information is modeled or compensated. But these results are shown to demonstrate the efficacy of the proposed controller and that it is sufficient to guarantee bipedal walking at a much-reduced DOFs of the robot legs.

The acceleration term, ${\ddot{\textbf{x}}}_R$ , in (6) is replaced by the control equation $\mathbf{u}$ such that the joints torque commands to the robot become,

(7) \begin{equation}\boldsymbol{\tau}_T = \mathbf{J}_R^T \mathbf{u} +(\mathbf{I} - \mathbf{J}_R^T \mathbf{J}_R^{+T}) \boldsymbol{\nabla} \mathbf{z} +\widetilde{\mathbf{g}}_T\end{equation}

where $\widetilde{\square}$ on top of a symbol representing a physical parameter means that it is the estimate of that parameter, $\mathbf{u}$ is the relative position control in the world space, and $\boldsymbol{\nabla} \mathbf{z}$ is the posture control. In this implementation, we only control the relative feet position ${}^2\mathbf{p}_4$ , with no control on relative feet orientation ${}^2\boldsymbol{\Phi}_4$ . Next, we will show how we control the world space relative feet position through $\mathbf{u}$ , and the corresponding posture to balance against gravity $\nabla\mathbf{z}$ .

4.1. World space relative position control

A simple PD control is used to control both the world space relative position, as well as the null-space posture control. The world space relative position control $\mathbf{u}$ from (7) can be expressed as

(8) \begin{equation}\mathbf{u} = - k_v\;{}^2{\dot{\textbf{p}}}_4 +k_p ({}^2\mathbf{p}_{4d} - {}^2\mathbf{p}_4)\end{equation}

where the value of the proportional gain $k_p=20$ and the value of the derivative gain $k_v=0.02$ . Subscript d means desired values. We compute the value of ${}^2\mathbf{p}_4$ from (3), and the translational velocity ${}^2{\dot{\textbf{p}}}_4$ is computed by

(9) \begin{equation}[{}^2{\dot{\textbf{p}}}_4,{}^2\boldsymbol{\omega}_4]= \mathbf{J}_R\left[\begin{matrix}{\dot{\textbf{q}}}_A^T & {\dot{\textbf{q}}}_B^T\end{matrix}\right]^T,\end{equation}

where $^2\boldsymbol{\omega}_4$ is the relative angular velocity, and ${\dot{\textbf{q}}}_A$ and ${\dot{\textbf{q}}}_B$ are the leg velocities. We set the desired values of the relative position by going through a cycle of alternately putting one foot ahead of the other, that is,

(10) \begin{equation}{}^2\mathbf{p}_{4d}=[(-1)^{c}0.2 a \;\;(-1)^{c}0.2\;\;0.25]^T\end{equation}

where $c\in{N}$ that increments every $t=0.5s$ and $a\in\{1,0\}$ switches between 1 and 0 every $0.5t$ . Note that Gazebo does not run in real time. Physically this means that at the start of the cycle, the left foot (of leg B) simultaneously lifts and moves forward (w.r.t. the right foot), then touches the ground. In the second half of the cycle, the right foot (of leg A) simultaneously lifts and moves forward (w.r.t. the left foot), then touches the ground.

4.2. Null-space control

Posture is controlled using the feedback of the IMUs placed at both ends of the hip link connecting the two legs as shown in Fig. 1. In this arrangement, using Fig. 1 frame assignment and labeling the ground as frame {0}, the IMU feedback are ${}^0\mathbf{R}_{1}$ and ${}^0\mathbf{R}_{3}$ from frame {1} and frame {3}, respectively. The desired orientation is set to be the orientation of the hip w.r.t. the ground at the initial stage of the walking, that is, ${}^0\mathbf{R}_{1d} = {}^0\mathbf{R}_{3d} = \mathbf{I}$ where $\mathbf{I}$ is the identity matrix. By considering the rotation matrix to be $\mathbf{R}=[\mathbf{n} \;\mathbf{o} \;\mathbf{a}]$ , a generic orientation control is

(11) \begin{equation}\begin{split}\mathbf{v}& = \frac{1}{2}\left[{}\mathbf{n}\times {}\mathbf{n}_{d} +{}\mathbf{o}\times {}\mathbf{o}_{d} +{}\mathbf{a}\times {}\mathbf{a}_{d}\right] \\\end{split}\end{equation}

such that the null-space controller becomes,

(12) \begin{equation}\begin{split}\mathbf{v}_1 = \frac{1}{2}\left[{}^1\mathbf{n}_0\times {}^1\mathbf{n}_{0d} +{}^1\mathbf{o}_0\times {}^1\mathbf{o}_{0d} +{}^1\mathbf{a}_0\times {}^1\mathbf{a}_{0d}\right] -k_w{}^1\omega_0\\\mathbf{v}_2 = \frac{1}{2}\left[{}^3\mathbf{n}_0\times {}^3\mathbf{n}_{0d} +{}^3\mathbf{o}_0\times {}^3\mathbf{o}_{0d} +{}^3\mathbf{a}_0\times {}^3\mathbf{a}_{0d}\right] -k_w{}^3\omega_0\end{split}\end{equation}

(13) \begin{equation}\nabla{z} =\left[\begin{matrix}{}^1\mathbf{J}_2^Tk_n\mathbf{v}_1 \\{}^3\mathbf{J}_4^Tk_n \mathbf{v}_2\end{matrix}\right]\end{equation}

where $k_n=10$ and $k_w=0.001$ . The angular velocities ${}^1\omega_0$ and ${}^3\omega_0$ are read directly from the IMUs.

5. Other cases considered

This section presents other cases of bipedal walking that can result in different walking scenarios, given the same robot as in Fig. 1 but using a different control approach.

5.1. Absolute foot movement

The case of absolute foot movement control is where a foot movement is based w.r.t. a hip frame. Using the bipedal robot in Fig. 1, this means that frame {2} moves w.r.t. frame {1} and frame {4} moves w.r.t. frame {3}. In this case, the corresponding absolute Jacobians (as shown above and we state here again for clarity) for leg A is $\mathbf{J}_A = {}^1\mathbf{J}_2\in\mathbb{R}^{3 \times 3}$ and for leg B is $\mathbf{J}_B = {}^3\mathbf{J}_4\in\mathbb{R}^{3 \times 3}$ . In this case, each leg is controlled individually, thus the total torques supplied to the joints will be different from (6), that is,

(14) \begin{equation}\boldsymbol{\tau}_T=\left[\begin{matrix}\boldsymbol{\tau}_A\\\boldsymbol{\tau}_B\end{matrix}\right]=\left[\begin{matrix}\mathbf{J}_A^T{\ddot{\textbf{x}}}_A\\\mathbf{J}_B^T{\ddot{\textbf{x}}}_B\end{matrix}\right]+\left[\begin{matrix}\mathbf{g}_A\\\mathbf{g}_B\end{matrix}\right]\end{equation}

where ${\ddot{\textbf{x}}}_A\in\mathbb{R}^3$ is the ${\ddot{\textbf{x}}}_B\in\mathbb{R}^3$ . And because each leg moves in the full-position space, the absolute Jacobians $\mathbf{J}_A$ and $\mathbf{J}_B$ are not redundant w.r.t. ${\ddot{\textbf{x}}}_A$ and ${\ddot{\textbf{x}}}_B$ , respectively. That is, all the DOFs of the each Jacobian is utilized to control the absolute foot movement and there is no DOF left to be used for balancing against gravity. Whereas, if relative feet movement is used, (6) will apply where $\mathbf{J}_R\in\mathbb{R}^{3 \times 6}$ is redundant w.r.t. ${\ddot{\textbf{x}}}_R\in\mathbb{R}^3$ . In this case, there will be extra DOFs to be used to balance against gravity.

Figure 2. Snapshots of the bipedal robot during the simulation experiment in Gazebo.

Figure 3. The relative feet position ${}^2\mathbf{p}_4$ plotted against the desired relative position ${}^2\mathbf{p}_{4d}$ .

Figure 4. Orientation error of the hip link w.r.t. the world coordinate, based on the feedback of the two IMUs placed at the link ends.

5.2. Relative feet movement with absolute position control

Relative foot movement with balancing against gravity indeed can perform a walking task for a bipedal robot without falling. But to give it a direction of motion w.r.t. the world frame, we would need to specify this additional task to the control equation. In order to achieve this, we modify (6) to become [Reference Jamisola and Roberts8,Reference Jamisola, Chang and Lee55],

(15) \begin{equation}\boldsymbol{\tau}_T =\mathbf{J}_R^T {\ddot{\textbf{x}}}_R+(\mathbf{I} - \mathbf{J}_R^T \mathbf{J}_R^{+T})\mathbf{J}_O^T\left[\begin{matrix}{}^1\dot{\omega}_0\\{}^3\dot{\omega}_0\end{matrix}\right]+(\mathbf{I} - \mathbf{J}_R^T \mathbf{J}_R^{+T})(\mathbf{I} - \mathbf{J}_O^T \mathbf{J}_O^{+T})\left[\begin{matrix}{}^0\mathbf{J}_2 & \mathbf{0}\end{matrix}\right]^T{}^0{\ddot{\textbf{x}}}_2+\mathbf{g}_T\end{equation}

where $\mathbf{J}_O =[\mathbf{J}_A\;\mathbf{0};\mathbf{0}\;\mathbf{J}_B]$ ; ${}^1\dot{\omega}_0,{}^3\dot{\omega}_0$ are the rotational acceleration of the world frame w.r.t. frame {1} and frame {3}, respectively; ${}^0\mathbf{J}_2 = {}^0\mathbf{R}_1 {}^1\mathbf{J}_2$ and ${}^0\ddot{\mathbf{x}}_2 = {}^0\mathbf{R}_1{}^1\ddot{\mathbf{x}}_2 = [{}^0\ddot{x}_2,{}^0\ddot{y}_2,0]^T$ . Counting the DOF needed for the allocated tasks, there are 3-DOFs for $\ddot{\mathbf{x}}_R$ , 2-DOFs for ${}^1\ddot{\mathbf{x}}_2$ , and the remaining 1-DOF for balancing against gravity. However, because balancing has higher priority than ${}^1\ddot{\mathbf{x}}_2$ , it can be possible that once $\ddot{\mathbf{x}}_R$ is performed, all the remaining 3-DOFs will be allocated for balancing before ${}^1\ddot{\mathbf{x}}_2$ is performed.