**2.2. Bipedal model**

To further investigate the effects of compliant joints and segmented feet on dynamic bipedal walking, in this section, we propose a passivity-based dynamic bipedal walking model that is more close to human beings. In the model, compliant ankle joints and flat segmented feet with compliant toe joints are included. As shown in Fig. 2, the two-dimensional model consists of two rigid legs interconnected individually through a hinge. Each leg contains segmented foot. The mass of the walker is divided into several point masses: hip mass, leg masses, masses of foot without toe, toe masses. Each point mass is placed at the center of corresponding stick. Torsional springs are mounted on both ankle joints and toe joints to represents joint stiffness.

**Figure 2.** Passivity-based dynamic bipedal walking model with flat segmented feet and compliant ankle joints.

To simplify the motion, we have several assumptions, including that legs suffering no flexible deformation, hip joint with no damping or friction, the friction between walker and ground is enough, thus the flat feet do not deform or slip, and strike is modeled as an instantaneous, fully inelastic impact where no slip and no bounce occurs. The passive walker travels on a flat slope with a small downhill angle.

The process of push-off is dissipated into foot rotation around toe joint and around toe tip, which is the main difference between the passive walking models with rigid flat feet and with segmented flat feet. The toe and foot are restricted into a straight line during the swing phase. When the flat foot strikes the ground, there are two impulses, "heel-strike" and "foot-strike", representative of the initial impact of the heel and the following impact as the whole foot contacts the ground. After foot-strike, the stance leg and the swing leg will be swapped and another walking cycle will begin. The passive walking is restricted to stop in two cases, including falling down and running. We deem that the walker falls down if the angle of either leg exceeds the normal range. The model is considered to running when the stance leg lifts up while the swing foot has not yet contacted the ground. Foot-scuffing at mid-stance is neglected since the model does not include knee joints.

We suppose that the x-axis (hereafter called horizontal coordinate) is along the slope while the y-axis (hereafter called vertical coordinate) is orthogonal to the slope and pointing upwards. The configuration of the walker is defined by the coordinates of the point mass on hip joint and six angles (swing angles between vertical coordinates and each leg, foot angles between horizontal coordinates and each foot, toe angles between horizontal coordinates and each toe), which can be arranged in a generalized vector *q* = (*xh*, *yh*, *α*1, *α*2, *α*<sup>1</sup> *<sup>f</sup>* , *α*<sup>2</sup> *<sup>f</sup>* , *α*1*t*, *α*2*t*)*<sup>T</sup>* (see Fig. 2). The positive direction of all the angles are counter-clockwise.

550 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges Segmented Foot with Compliant Actuators and Its Applications to Lower-Limb Prostheses and Exoskeletons <sup>5</sup> Segmented Foot with Compliant Actuators and Its Applications to Lower-Limb Prostheses and Exoskeletons 551

#### **2.3. Walking dynamics**

4 Will-be-set-by-IN-TECH

**Figure 2.** Passivity-based dynamic bipedal walking model with flat segmented feet and compliant ankle

To simplify the motion, we have several assumptions, including that legs suffering no flexible deformation, hip joint with no damping or friction, the friction between walker and ground is enough, thus the flat feet do not deform or slip, and strike is modeled as an instantaneous, fully inelastic impact where no slip and no bounce occurs. The passive walker travels on a flat

The process of push-off is dissipated into foot rotation around toe joint and around toe tip, which is the main difference between the passive walking models with rigid flat feet and with segmented flat feet. The toe and foot are restricted into a straight line during the swing phase. When the flat foot strikes the ground, there are two impulses, "heel-strike" and "foot-strike", representative of the initial impact of the heel and the following impact as the whole foot contacts the ground. After foot-strike, the stance leg and the swing leg will be swapped and another walking cycle will begin. The passive walking is restricted to stop in two cases, including falling down and running. We deem that the walker falls down if the angle of either leg exceeds the normal range. The model is considered to running when the stance leg lifts up while the swing foot has not yet contacted the ground. Foot-scuffing at mid-stance is

We suppose that the x-axis (hereafter called horizontal coordinate) is along the slope while the y-axis (hereafter called vertical coordinate) is orthogonal to the slope and pointing upwards. The configuration of the walker is defined by the coordinates of the point mass on hip joint and six angles (swing angles between vertical coordinates and each leg, foot angles between horizontal coordinates and each foot, toe angles between horizontal coordinates and each toe), which can be arranged in a generalized vector *q* = (*xh*, *yh*, *α*1, *α*2, *α*<sup>1</sup> *<sup>f</sup>* , *α*<sup>2</sup> *<sup>f</sup>* , *α*1*t*, *α*2*t*)*<sup>T</sup>* (see Fig.

joints.

slope with a small downhill angle.

neglected since the model does not include knee joints.

2). The positive direction of all the angles are counter-clockwise.

The model can be defined by the rectangular coordinates *r*, which can be described by the x-coordinate and y-coordinate of the mass points and the corresponding angles (suppose leg 1 is the stance leg):

$$\mathbf{r} = \begin{bmatrix} \mathbf{x}\_{\mathsf{h}} \ y\_{\mathsf{h}}, \mathbf{x}\_{\mathsf{c1}}, \mathbf{y}\_{\mathsf{c1}}, \mathbf{z}\_{\mathsf{c2}}, \mathbf{y}\_{\mathsf{c2}}, \mathbf{z}\_{\mathsf{c2}}, \mathbf{z}\_{\mathsf{c1}f}, \mathbf{y}\_{\mathsf{c1}f}, \mathbf{z}\_{\mathsf{c1}f}, \mathbf{z}\_{\mathsf{c2}f}, \mathbf{z}\_{\mathsf{c2}f}, \mathbf{z}\_{\mathsf{c1}s}, \mathbf{y}\_{\mathsf{c1}s}, \mathbf{z}\_{\mathsf{c2}s}, \mathbf{y}\_{\mathsf{c2}s}, \mathbf{z}\_{\mathsf{c2}s} \end{bmatrix}^{\mathsf{T}} \text{(1)}$$

The walker can also be described by the generalized coordinates *q* as mentioned before:

$$q = [\mathfrak{x}\_{\text{h}}, y\_{\text{h}}, \mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \mathfrak{a}\_{1f}, \mathfrak{a}\_{2f}, \mathfrak{a}\_{1t}, \mathfrak{a}\_{2t}]^{\text{T}} \tag{2}$$

The definitions of variables mentioned above can be found in Fig. 2.

We defined matrix *T* as follows:

$$T = \frac{dr}{dq} \tag{3}$$

Thus *T* transfers the independent generalized coordinates *q*˙ into the velocities of the rectangular coordinates *r*˙. The mass matrix in rectangular coordinates *r* is defined as:

$$M = \text{diag}\left(m\_{\text{Ir}}, m\_{\text{Ir}}, m\_{\text{I}}, m\_{\text{I}}, \text{I}\_{\text{I}}, m\_{\text{I}}, m\_{\text{I}}, \text{I}\_{\text{I}}, m\_{\text{J}} - m\_{\text{S}}, m\_{\text{J}} - m\_{\text{S}}, \text{I}\_{\text{J}}, m\_{\text{J}} - m\_{\text{S}}\right)$$

$$m\_{\text{J}} - m\_{\text{S}}, \text{I}\_{\text{J}}, m\_{\text{S}}, m\_{\text{S}}, \text{I}\_{\text{S}}, m\_{\text{S}}, m\_{\text{S}}, \text{I}\_{\text{S}}\right) \tag{4}$$

Denote *F* as the active external force vector in rectangular coordinates. The constraint function, which is used to maintain foot contact with ground and detect impacts, is marked as *ξ*(*q*). Note that *ξ*(*q*) in different walking phases may be different since the contact conditions change.

We can obtain the Equation of Motion (EoM) by Lagrange's equation of the first kind:

$$M\_{\boldsymbol{\theta}}\ddot{\boldsymbol{q}} = F\_{\boldsymbol{\theta}} + \boldsymbol{\Phi}^{\boldsymbol{T}}F\_{\boldsymbol{\xi}} \tag{5}$$

where *Fc* is the contact force acted on the walker by the ground to meet the constraint of the stance foot.

$$
\xi(q) = 0 \tag{6}
$$

where Φ = *∂ξ <sup>∂</sup><sup>q</sup>* . *Mq* is the mass matrix in the generalized coordinates:

$$M\_q = T^T M T \tag{7}$$

*Fq* is the active external force in the generalized coordinates:

$$F\_q = T^T F - T^T M \frac{\partial T}{\partial q} \dot{q} \dot{q} \tag{8}$$

Equation (6) can be transformed to the following equation:

$$
\Phi \ddot{\eta} = -\frac{\partial (\Phi \dot{\eta})}{\partial q} \dot{q} \tag{9}
$$

#### 6 Will-be-set-by-IN-TECH 552 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Then the EoM in matrix format can be obtained from Equation (5) and Equation (9):

$$
\begin{bmatrix} M\_q - \Phi^T \\ \Phi \end{bmatrix} \begin{bmatrix} \ddot{q} \\ F\_c \end{bmatrix} = \begin{bmatrix} F\_q \\ -\frac{\partial(\Phi \dot{q})}{\partial q} \dot{q} \end{bmatrix} \tag{10}
$$

The equation of strike moment can be obtained by integration of Equation (5):

$$M\_{\boldsymbol{q}}\dot{\boldsymbol{q}}^{+} = M\_{\boldsymbol{q}}\dot{\boldsymbol{q}}^{-} + \boldsymbol{\Phi}^{T}\boldsymbol{I}\_{\boldsymbol{c}} \tag{11}$$

where *q*˙ <sup>+</sup> and *q*˙ − are the velocities of generalized coordinates after and before the strike, respectively. Here, *Ic* is the impulse acted on the walker which is defined as follows:

$$I\_{\mathcal{C}} = \lim\_{t^- \to t^+} \int\_{t^-}^{t^+} F\_{\mathcal{C}} dt \tag{12}$$

where *Ic* is the impulse acted on the walker by ground. Since the strike is modeled as a fully inelastic impact, the walker satisfies the constraint function *ξ*(*q*). Thus the motion is constrained by the following equation after the strike:

$$i\frac{\partial \xi}{\partial q}\dot{q}^+ = 0\tag{13}$$

Then the equation of strike in matrix format can be derived from Equation (11) and Equation (13):

$$
\begin{bmatrix} M\_{\dot{q}} - \Phi^T \\ \Phi \end{bmatrix} \begin{bmatrix} \dot{q}^+ \\ I\_c \end{bmatrix} = \begin{bmatrix} M\_{\dot{q}} \dot{q}^- \\ 0 \end{bmatrix} \tag{14}
$$

More detailed description of the bipedal model can be found in [24].

#### **3. Effects of compliant joints and segmented feet**

Based on the EoMs mentioned above, we analyze the effects of compliant ankles and toes on energetic efficiency and stability of dynamic bipedal walking. All masses and lengthes are normalized by the leg mass and leg length respectively. The spring constants (stiffness of ankle joint and toe joint) are normalized by both the mass and the length of leg.

#### **3.1. Energetic efficiency**

Energetic efficiency is an important gait characteristics. The energy consumption of passive dynamic based models is usually represented in the nondimensional form of "specific resistance": energy consumption per kilogram mass per distance traveled per gravity [8]. However, for passive walkers on a gentle slope, specific resistance is not a suitable measure of efficiency, since all walkers have the same specific resistance for a given slope [12]. Therefore, similar to [12], normalized walking velocity is used as the measure of efficiency, such that "most efficient" is synonymous of "fastest".

The walking velocity of the rigid foot model (without toe joints) decreases monotonically as foot length or foot ratio (the ratio of distance between heel and ankle joint to distance between ankle joint and toe tip) grows (see Fig. 3(a)). For the segmented foot model, the walker moves slower for longer foot according to the main tendency (see Fig. 3(b)). Walking velocity achieves the maximum value when foot ratio is near 0.3 for any fixed foot length.

**Figure 3.** Comparison of walking velocities of rigid foot model and segmented foot model. The curved surfaces are smooth processed based on the sample data. (a) Average walking velocity versus foot length and foot ratio for rigid flat-foot passive walking model. (b) Average walking velocity versus foot length and foot ratio for segmented flat-foot passive walking model. (c) The difference of walking velocity of the two models, obtained from (b) subtracted by (a). Both walking velocity and foot length are normalized by leg length. Foot ratio is defined as the ratio of distance between heel and ankle joint to distance between ankle joint and toe tip.

A peak appears at relative large foot length (larger than 0.2) and foot ratio near 0.3, which is similar to the foot structure of human beings [25]. The comparison of the two models shows that the walker with segmented feet moves slower than rigid foot model with small foot length, however, the velocity of segmented foot model is larger when the foot is long enough, especially when foot ratio is near 0.3. In another word, if the segmented foot ratio is close real human foot, the segmented foot model is more efficient than the rigid foot model.

## **3.2. Walking stability**

6 Will-be-set-by-IN-TECH

 *Fq* −*∂*(Φ*q*˙) *<sup>∂</sup><sup>q</sup> q*˙

− are the velocities of generalized coordinates after and before the strike,

 *t* +

where *Ic* is the impulse acted on the walker by ground. Since the strike is modeled as a fully inelastic impact, the walker satisfies the constraint function *ξ*(*q*). Thus the motion is

Then the equation of strike in matrix format can be derived from Equation (11) and Equation

Based on the EoMs mentioned above, we analyze the effects of compliant ankles and toes on energetic efficiency and stability of dynamic bipedal walking. All masses and lengthes are normalized by the leg mass and leg length respectively. The spring constants (stiffness of

Energetic efficiency is an important gait characteristics. The energy consumption of passive dynamic based models is usually represented in the nondimensional form of "specific resistance": energy consumption per kilogram mass per distance traveled per gravity [8]. However, for passive walkers on a gentle slope, specific resistance is not a suitable measure of efficiency, since all walkers have the same specific resistance for a given slope [12]. Therefore, similar to [12], normalized walking velocity is used as the measure of efficiency, such that

The walking velocity of the rigid foot model (without toe joints) decreases monotonically as foot length or foot ratio (the ratio of distance between heel and ankle joint to distance between ankle joint and toe tip) grows (see Fig. 3(a)). For the segmented foot model, the walker moves slower for longer foot according to the main tendency (see Fig. 3(b)). Walking velocity achieves the maximum value when foot ratio is near 0.3 for any fixed foot length.

 *q*˙ + *Ic* = *Mqq*˙ − 0

<sup>−</sup> + Φ*<sup>T</sup> Ic* (11)

*<sup>t</sup>*<sup>−</sup> *Fcdt* (12)

<sup>+</sup> = 0 (13)

(10)

(14)

Then the EoM in matrix format can be obtained from Equation (5) and Equation (9):

 *q*¨ *Fc* =

<sup>+</sup> = *Mqq*˙

respectively. Here, *Ic* is the impulse acted on the walker which is defined as follows:

*Ic* = lim *<sup>t</sup>*−→*t*<sup>+</sup>

> *∂ξ ∂q q*˙

constrained by the following equation after the strike:

**3. Effects of compliant joints and segmented feet**

More detailed description of the bipedal model can be found in [24].

*Mq* <sup>−</sup>Φ*<sup>T</sup>* Φ 0

ankle joint and toe joint) are normalized by both the mass and the length of leg.

where *q*˙

(13):

<sup>+</sup> and *q*˙

**3.1. Energetic efficiency**

"most efficient" is synonymous of "fastest".

*Mq* <sup>−</sup>Φ*<sup>T</sup>* Φ 0

The equation of strike moment can be obtained by integration of Equation (5):

*Mqq*˙

We evaluate adaptive walking of the model on uneven terrain to analyze the walking stability. Figure 4 shows the relationship between the maximal allowable ground disturbance (a step down) the walker can overcome and the foot ratio of rigid foot bipedal model. The maximal allowable ground disturbance decreases monotonically as the foot ratio grows, which is

**Figure 4.** Adaptability of the rigid foot model with different foot ratios. The normalized foot length is 0.1875. The ground disturbance is also normalized by leg length.

similar to the trend of walking velocity. In case of short hindfoot and long forefoot (foot ratio is 0.2), the walker can return to stable motion cycle after a ground disturbance larger than 0.7 percent of leg length. However, the maximum disturbance the model can overcome decreases below 0.2 percent of leg length when the lengths of hindfoot and forefoot are comparable (foot ratio is 0.8). The relationship between the maximal allowable ground disturbance and foot ratio of segmented foot model also shows a great resemblance to the trend of walking velocity (Fig. 5 shows the results). The maximum value is obtained when the foot ratio is 0.3.

**Figure 5.** Adaptability of the segmented foot model with different foot ratios. The normalized foot length is 0.1875. The ground disturbance is also normalized by leg length.

In that case the model can overcome ground disturbance larger than 0.6 percent of leg length. The adaptability of the segmented foot model decreases significantly if the foot ratio changes to other values. The results indicate that there exists a best foot structure of the segmented foot model, which achieves both excellent adaptability and walking velocity.

From the analysis above, one can find that the segmented foot model has comparable walking adaptability with rigid foot models in the case of suitable foot ratios. However, the walkers with segmented feet perform worse in other cases.
