A Novel Multi-Gait Strategy for Stable and Efficient
Quadruped Robot Locomotion

Quadruped animals lift their legs and touch down in a coordinated cyclic manner, known as gait, to achieve balance and efficient locomotion [26]. Inspired by natural quadrupeds, we also define five gait patterns for quadruped robots and implement them on flat and slope terrains.

The gait patterns can be characterized as the representation of gait parameters, duty factor $\beta$ and the four legs’ phase offsets $\boldsymbol{\phi}$ defined as $(\phi_{RF},\phi_{RH},\phi_{LF},\phi_{LH})$ which represent the swing phase offset in the right front leg, right hind leg, left front leg and left hind leg, respectively. The duty factor $\beta$ represents the ratio of one leg stance duration to one stride period, and each leg’s phase offset $\phi$ is defined as the lift-off (LO) timing in one stride period:

		$\displaystyle\beta=\frac{T_{st}}{T_{st}+T_{sw}}$		(1)
		$\displaystyle\phi=\frac{t_{LO}}{T_{st}+T_{sw}}$

where $T_{st}$ represents the stance duration and $T_{sw}$ represents the swing duration; the sum of them is a stride period. All five gait patterns used in this work are illustrated as a phase diagram shown in Fig. 2.

Walking is a static gait for its duty factor $\beta>0.5$, which means at least three feet of contact with the ground at each moment. For trot and bound gait, their duty factor $\beta$ is 0.5, which represents keeping at least two contact points in one stride period without the flight phase (all feet are lifted off the ground). The difference between trot and bound gait is that the former’s contact points are at a diagonal line while the other is at the front or hind of the robot. Therefore, trotting has less vibration than bounding when the robot starts moving. Trot-run and run gait are the upgrade modes for trotting and bounding with a flight phase. Since their duty factors $\beta=0.3$ are less than 0.5, both can be considered dynamic gaits and perform better in energy efficiency than their original mode, as demonstrated in Sec. IV.

The gait patterns are discrete for their different $\beta$ and $\boldsymbol{\phi}$, so it is hard to transform one gait into another instantaneously considering the physical limits. However, some gait patterns share the same parameters, making the transition between them much easier. For instance, trotting and bounding have the same $\beta$ and differ at $\boldsymbol{\phi}$. Therefore, we segment the continuous transition among all the gait patterns into four switching processes based on similarity, which are walk$\leftrightarrow$trot, trot$\leftrightarrow$bound, trot$\leftrightarrow$trot-run, and bound$\leftrightarrow$run. After that, the switching process between walk and trot can be built as follows:

		$\displaystyle\beta(t)=\alpha_{0}\oplus\frac{1}{4T_{s}}t,t\in[0,T_{s}]$		(2)
		$\displaystyle\phi_{LF}=0.0,\phi_{LH}=0.5,\phi_{RF}=\beta(t),\phi_{RH}=\beta(t)\oplus 0.5$
		$\displaystyle trot\rightarrow walk:\oplus=+,\alpha_{0}=0.5$
		$\displaystyle walk\rightarrow trot:\oplus=-,\alpha_{0}=0.75$

In the same way, another three switching processes can be characterized as follows, where trot $\leftrightarrow$ bound is:

		$\displaystyle\beta(t)=0.5,t\in[0,T_{s}]$		(3)
		$\displaystyle\phi_{LF}=0.0,\phi_{LH}=0.5$
		$\displaystyle trot\rightarrow bound:\phi_{RF}=0.5-\frac{1}{2T_{s}}t,\phi_{RH}=\frac{1}{2T_{s}}t$
		$\displaystyle bound\rightarrow trot:\phi_{RF}=\frac{1}{2T_{s}}t,\phi_{RH}=0.5-\frac{1}{2T_{s}}t$

trot $\leftrightarrow$ trot-run is:

		$\displaystyle\beta(t)=\alpha_{0}\oplus\frac{1}{5T_{s}}t,t\in[0,T_{s}]$		(4)
		$\displaystyle\phi_{LF}=0.0,\phi_{LH}=0.5,\phi_{RF}=0.5,\phi_{RH}=0.0$
		$\displaystyle trot\rightarrow trot-run:\oplus=-,\alpha_{0}=0.5$
		$\displaystyle trot-run\rightarrow trot:\oplus=+,\alpha_{0}=0.3$

bound $\leftrightarrow$ run is:

		$\displaystyle\beta(t)=\alpha_{0}\oplus\frac{1}{5T_{s}}t,t\in[0,T_{s}]$		(5)
		$\displaystyle\phi_{LF}=0.0,\phi_{LH}=0.5,\phi_{RF}=0.0,\phi_{RH}=0.5$
		$\displaystyle bound\rightarrow run:\oplus=-,\alpha_{0}=0.5$
		$\displaystyle run\rightarrow bound:\oplus=+,\alpha_{0}=0.3$

where $T_{s}$ is the transition time for each gait, and $\oplus$ is an operator representing $+$ or $-$ according to the transition. All the switching between two similar gaits can be accomplished during the transition time. Considering trot-run and run are both dynamic gaits with high instability, the direct transition between them will be denied, for it easily causes the robot to fall.

Locomotion controllers for legged robots need to manage the continuous dynamics of the system and the discrete gait transition. For this reason, we design a graph to define the switching action between gaits, as shown in Fig. 3. Furthermore, a finite state machine (FSM) is developed to set the switching order among all the gait patterns, and we define the related state numbers for different gaits: walking, trotting, bounding, running, and trot-running are represented as 0, 1, 2, 3, 4, respectively. The events that realize transitions between states are achieved through no more than three actions, as shown in Tab. I.

Achieving efficient and stable locomotion for quadruped robots depends on the gait transition and determining an appropriate gait for different terrains. Therefore, selecting a suitable gait to adapt to various terrains is necessary and valuable.

Firstly, the energy consumption during locomotion needs to be figured out. The total power is the sum of the mechanical motor power and the motor copper losses of all joints. Since our experiments are conducted on the quad_sdk [11], a quadruped robot simulator based on ROS and Gazebo, the robot will not produce heat losses and cannot store excess electrical energy in batteries or capacitors. If all motors create negative power, this power should be considered dissipating in the system. Then, the consumption energy $W$ can be integrated over a full stride as follows [6]:

$$W=\int_{0}^{t_{f}}(\sum_{i=1}^{n}max(u_{i}\cdot\omega_{i},0))dt$$

(6)

To compare energetic economy across different terrains with various velocities, the cost of transport (CoT) is introduced:

$$\text{CoT}=\frac{W}{mg\Delta s}$$

(7)

where $t_{f}$ represents one stride period, $n=12$ is the motor numbers of the robot, $u_{i}$ and $\omega_{i}$ represent the torque and the angular velocity of each motor respectively, $m$ is the total mass of the robot and $\Delta s$ represents the travel distance in one stride period.

Furthermore, the robot should remain stable enough to guarantee basic locomotion in different environments. Hence, we build an index to express the robot’s stability under locomotion called STB, which is a variant of the $J_{3}$ in [5].

$$\text{STB}=w_{1}(|v_{bn}/v_{b}|)+w_{2}|\theta_{b}-\theta_{t}|+w_{3}|\phi_{b}|+w_{4}(|\dot{\theta}_{b}|+|\dot{\phi}_{b}|))$$

(8)

where $v_{bn}$ represents the normal velocity of the robot, $v_{b}$ is the robot’s actual velocity, $\theta_{b}$ and $\phi_{b}$ represent the body’s pitch and roll angle, respectively. $\theta_{t}$ is the inclination of the terrain and $\dot{\theta}_{b}$, $\dot{\phi}_{b}$ are the attitude change rate. STB integrates the robot’s attitude variation and vertical vibration to represent body stability. Apparently, the unstable locomotion is mainly caused by the robot’s attitude, so the weight of attitude terms $w_{2},w_{3}$ are set as 1.0, $w_{1}$ and $w_{4}$ are set as 0.7 and 0.3.

Both CoT and STB will be affected by gait patterns $\boldsymbol{\Lambda}$ and should be considered simultaneously. Although the CoT and STB have different dimensions, they share a similar scale during normal locomotion (from 0 to 1.4, which can be seen in Fig. 5). So, the two indexes are integrated into one total index $J_{e}(\boldsymbol{\Lambda})$:

$$J_{e}(\boldsymbol{\Lambda})=c\cdot\text{STB}+(1-c)\cdot\text{CoT}$$

(9)

where $c$ is the ratio of STB in the evaluation equation and can be altered to achieve different goals.

After that, the most suitable gait $\boldsymbol{\Lambda}^{*}$ for different terrains can be acquired by minimizing the $J_{e}$ while satisfying the robot’s dynamics and other constraints. Therefore, the optimization formulation can be constructed as follows:

		$\displaystyle min\quad(\sum_{i=1}^{N}J_{e}(\boldsymbol{\Lambda}))/N$		(10a)
	$\displaystyle s.t.\quad\quad\quad$	$\displaystyle m\ddot{\boldsymbol{r}}=m\boldsymbol{g}+\sum_{l=1}^{4}\boldsymbol{f}_{l}$
		$\displaystyle I\boldsymbol{\dot{\omega}}+\boldsymbol{\omega}\times(I\omega)=\sum_{l=1}^{4}\boldsymbol{f}_{l}\times(\boldsymbol{r}-\boldsymbol{p}_{l})$		(10b)
		$\displaystyle-\mu\boldsymbol{f}_{l}^{n}\leq\boldsymbol{f}_{l}^{t}\leq\mu\boldsymbol{f}_{l}^{n}\quad\quad stance\quad leg$		(10c)
		$\displaystyle\boldsymbol{I}\boldsymbol{f}_{l}=\boldsymbol{0}\quad\quad\quad\quad\quad\quad\quad swing\quad leg$		(10d)

where Eq. (10a) is the optimization goal, and $N$ is the pre-defined strides’ count. Eq. (10b) represents the dynamics constraints, where $\boldsymbol{r}\in\mathbb{R}^{3}$, $\boldsymbol{\omega}\in\mathbb{R}^{3}$ are the Center of Mass (CoM) position and angular velocity, and $\boldsymbol{p}_{l}\in\mathbb{R}^{3}$, $\boldsymbol{f}_{l}\in\mathbb{R}^{3}$represent the foot location and contact forces of leg $l$ expressed in the world frame. Eq. (10c) is the friction cone that prevents the slipping of the robot’s feet, where $\mu$ is the friction coefficient, $\boldsymbol{f}_{l}^{n}$ represents the normal component of the contact force, and $\boldsymbol{f}_{l}^{t}$ is the tangential component. $\boldsymbol{I}\in\mathbb{R}^{3\times 3}$ is the identity matrix and $\boldsymbol{0}\in\mathbb{R}^{3}$.

Extensive experiments are conducted to find the optimal gait on each terrain and ratio $c$ with various velocities. Then, we can transform these data into a velocity-gait map on each terrain by calculating Eq. (10a). Finally, the gait selection is completed by querying the map on any related terrains.

The first experiment is to evaluate the performance of the proposed transition method and its outcomes to support the claim that we realize the smooth transition among five distinct gait patterns and achieve the highly dynamic gait locomotion ( running gait and trot-running gait).

The proposed gait transition method is tested on flat terrain with transition duration $T_{s}$ set to 0.5 s, and the results are shown in Fig. 4. Fig. 4 (a) shows the transition from trotting to walking and trot-running with the robot’s speed is 1.2 m/s. At the beginning of Fig. 4 (a), the RF, HL curves are overlapped, and the same for the LF, RH curves, which represent the robot using the trotting gait. After that, the robot switches its gait to walking (from 8.5 s to 10.5 s) and to trot-running (after 14 s). In the walking gait part, there are four distinct peaks, representing that only one foot is in the air during each moment. In trot-running gait, the LF and RH curves overlap while the four curves intersect at the same point whose ordinate is higher than 0.02 m (the ground clearance), representing all feet in the air (the whole robot is in the flight phase). This result shows that the robot can maintain balance and attitude in a narrow area under trot-running gait, which verifies our first claim.

Fig. 4 (b) shows the transition from trotting to bounding and running at 1.9 m/s. The RF, LF curves are overlapped and the same for the LH, RH curves, which represent the robot using the bounding gait from 8 s to 10 s. After that, the robot switches its gait to trotting (from 10.5 s to 12s) and running (after 13 s). In the running gait part, the LF and RF curves still overlap while the four curves intersect at a high point, representing the whole robot in the flight phase. Additionally, the pitch angle curve represents that the robot has periodical vibration while still maintaining balance when taking the running gait. This illustrates that our method can achieve highly dynamic gait locomotion.

Beyond that, we can find that bounding and running cause more body vibration compared to other gaits. Apparently, trotting performs the most stable among the three gaits according to the variation of body attitude in the upper plot of Fig. 4 (a), which is due to its diagonal stepping without flying phase.

The second experiment explains the construction of the velocity-gait mapping and illustrates that our approach is capable of achieving the most suitable gait for quadruped robots on different terrains with velocities varying. To form the velocity-gait mapping, we collect the CoT and STB indexes with various velocities from 0.3 m/s to 2.7 m/s on flat and slope terrains, respectively. Each test with a fixed velocity on one terrain is repeated at least five times, and the curves of the average results are plotted in Fig. 5.

Fig. 5 (a) illustrates that trotting has the lowest STB on flat terrain while running shows higher energy efficiency with lower CoT when the robot’s velocity is faster than 0.7 m/s. Fig. 5 (b) demonstrates that trotting is the most stable gait with a velocity less than 2.3 m/s, and trot-running saves more energy when the robot moves at high speed on a slope. Besides, when the robot takes locomotion on a terrain, it may inevitably fall at some speed because of the controller’s limitation. If the robot falls down at each test, the correct CoT and STB values can not be collected. Therefore, the CoT and STB upper bounds are set as 1.25 and 1.36, respectively, near the maximum values in all normal tests. Furthermore, it is unreasonable to decide a gait for different velocities with only one index value. Therefore, combining two indexes according to Eq. (9) and Eq. (10) would be necessary, then we transform the merged data into a mapping from locomotion velocities to gait patterns. The results are shown in Fig. 6.

On a flat, trotting is the most proper gait on each $J_{e}$ index when the robot is at low speeds (up until 0.9 m/s). However, if we consider the energy efficiency (c is no more than 0.3) at middle or high speeds (more than 1.1 m/s). The model favors a running gait, which has been demonstrated by many research [4, 6, 15] as the best gait for saving energy in the natural quadrupeds with high speeds. Fig. 6 (b) shows that trot-running performs better on any $J_{e}$ index when the robot takes high-speed locomotion on a slope. The reason is that the trot-running gait can use diagonal stepping to achieve more stable locomotion than running and reduce the body’s swing on pitch direction, especially for a slope. At the same time, trot-running saves more energy by adding the flying phase compared to trotting. Therefore, trotting and trot-running are the most suitable for locomotion on slope terrain. Additionally, the result shown in Fig. 6 (b) coincides with the paper [5], whose results also illustrate that trot-running is the optimal gait at high speeds (more than 1.7 m/s) to maintain the robot’s locomotion stability in most situations. In a word, trotting is the best gait when the robot is at low speeds, whether on a flat or slope. Running gait is more suitable for saving energy when moving on a flat at high speed, while trot-running is better on a slope.

The last experiment is conducted to verify the efficiency of the proposed multi-gait strategy and demonstrate the practical applicability of the gait selection theory from biology to quadrupedal robots. The experimental process and related gaits are shown in Fig. 7 (a). The experimental setup is as follows: the robot receives a speed command, which forces it to move forward; our strategy will select the proper gait guided by the gait mapping on the current terrain and accomplish the gait transition until it reaches another terrain. At the same time, we introduce other methods, including two fixed gait methods which take the same gait for each test, and one multi-gait method which determines the optimal gait according to the velocity and does not switch gait during one test (Wang’s method [5]) as baselines, then record the corresponding data for comparison. For our strategy, we take the stability coefficient $c$ from 0.1 to 0.9 in the gait mapping when the robot moves on the whole terrain; each method is tested and repeated 30 times on a random velocity from 0.3 m/s to 2.7 m/s with the average at 1.481 m/s, the results are shown in Tab.  II.

All the results are calculated by introducing the failed effect, which means if the robot falls on the test ( can not accomplish the entire process), the CoT and STB values are set as the maximum, 1.25 and 1.36, respectively. According to the results, our strategy ($c=0.1$) shows the lowest energy consumption on the whole terrain among all methods, with only one failure. This strategy makes the robot run on flat terrain at high speed to improve energy utilization while trot-running on a slope to maintain stability. The trot-running method shows only a 76.7 % (23 /30) success rate for its unstable locomotion on the flat at low velocities, as shown in Fig. 5. Compared with other methods, our strategies with $c=0.5,0.7,0.9$ reach a similar STB at the lowest level, and their success rates are 100 %, meaning our strategies can achieve the most stable locomotion on the whole terrain. At the same time, the proposed strategy c5 shares a similar CoT with Wang’s method while reaching a lower STB than the latter. Therefore, our strategy c5 is the most compromised method to achieve stable and efficient locomotion among all the methods. Those results indicate that our multi-gait strategy enables the robot to achieve efficient and stable locomotion in the specified environment by selecting the proper gait and completing the transition. Additionally, the proposed method is also tested on some more complex terrains, which are the continuous slope (the inclinations are 8^∘, 12^∘ and 18^∘) and the up-down slope (both the inclinations are 12^∘). The results are shown in Fig. 7 (b) and Fig. 7 (c).

In the simulation test, the proposed multi-gait motion strategy achieves the desired motion effect. In order to verify that the multi-gait strategy can also be useful in real life, this paper carries out an outdoor experiment based on a Unitree go2 robotic dog, adopting the c1 strategy with a movement speed of 1.7 m/s, and the test environment is a flat-slope terrain, and the movement process is shown in Fig. 8. It can be seen that the multi-gait locomotion strategy makes the robot dog adopt running gait on flat ground and trot-run gait on slope, which is in line with the c1 strategy, and also agrees with the effect achieved in the simulation experiment. The motion data of the go2 robot was collected and the values of CoT and STB were plotted as shown in Fig. 9. It can be seen that, after adopting the multi-gait motion strategy, although the switch from flat to sloping terrain makes the energy consumption of the robot dog rise slightly (overcoming gravity to do work on the slope inevitably leads to a rise in energy consumption), it significantly reduces the value of the STB and ensures the stability of the robot dog’s motion on the slope, which proves that the proposed multi-gait locomotion strategy can be applied in practice and has the ability to balance the energy efficiency and stability of locomotion.