3.1.1 Planar motion of a rigid body

Consider the rigid movement of a planar body, $M$ , relative to a fixed body $F$ , defined by the rotation angle $\phi$ measured between the $x$ axes of $M$ and $F$ and the translation vector $\mathbf{d}=(a, b)$ measured from the origin of $F$ to the origin of $M$ . If the parameters $\phi$ , $a$ and $b$ vary with time, then planar motion is the parameterized transformation of coordinates $\mathbf{p}$ of a point in $M$ to its trajectory $\mathbf{P}(t)$ in $F$ , given by,

(1) \begin{equation} \mathbf{P}(t)=\begin{Bmatrix} X(t) \\ Y(t) \\1 \end{Bmatrix}=\begin{bmatrix} \cos \phi (t) & -\sin \phi (t) & a(t) \\ \sin \phi (t) & \cos \phi (t) & b(t) \\ 0 & 0 & 1\end{bmatrix}\begin{Bmatrix}x \\ y \\ 1\end{Bmatrix}= \mathbf{D}(t)\mathbf{p}. \end{equation}

The $3\times 3$ homogeneous transform $\mathbf{D}$ is similar to the $4\times 4$ homogeneous transforms used in Robotics, McCarthy [Reference McCarthy31]. We use this $3\times 3$ matrix to simplify our derivations, however, we will drop the homogeneous coordinate 1 in $\mathbf{P}(t)$ and $\mathbf{p}$ when it is convenient.

The focus of curvature theory is on the geometric properties of point trajectories, therefore, rather than using time $t$ as a parameter, we use the rotation angle $\phi$ of the moving body. We reparameterize the components of the translation vector so we have $a(\phi )$ and $b(\phi )$ . Thus, the motion of a planar rigid body is given by the coordinate transformation,

(2) \begin{equation} \mathbf{P}(\phi )=\begin{Bmatrix} X(\phi ) \\ Y(\phi ) \\1 \end{Bmatrix}=\begin{bmatrix} \cos \phi & -\sin \phi & a(\phi ) \\ \sin \phi & \cos \phi & b(\phi ) \\ 0 & 0 & 1\end{bmatrix}\begin{Bmatrix}x \\ y \\ 1\end{Bmatrix}= \mathbf{D}(\phi )\mathbf{p}. \end{equation}

This equation defines the trajectory $\mathbf{P}(\phi )$ traced in a fixed frame $F$ by the point $\mathbf{p}$ in the moving frame $M$ . This can be viewed as a map of all the points of $M$ to trajectories in $F$ .

In order to study the differential properties of a trajectory $\mathbf{P}(\phi )$ , we introduce notation for the derivatives of $\mathbf{P}$ at the instant $\phi =0$ ,

(3) \begin{equation} \mathbf{P}(0) = \mathbf{P}_0,\quad \frac{d\mathbf{P}}{d\phi }(0) = \mathbf{P}_1, \quad \frac{d^2\mathbf{P}}{d\phi ^2}(0) = \mathbf{P}_2, \quad \frac{d^3\mathbf{P}}{d\phi ^3}(0) = \mathbf{P}_3, \quad \ldots \,\,. \end{equation}

We use this notation in the definition of the instantaneous invariants of a planar motion.

3.1.2 Instantaneous invariants

In every position of the moving body $M$ , there is a point $\mathbf{V}$ , known as the velocity pole or the instant center of rotation, that has zero velocity relative to the fixed body $F$ . The set of poles for the movement of $M$ relative to $F$ is called a centrode. For a specific position of the moving body, we choose the pole $\mathbf{V}$ as the origin of coordinates for both the fixed frame $F$ and moving frame $M$ , and the tangent to the centrode as the x-axis for what is known as the canonical coordinate system.

Using the canonical coordinate system, we obtain the Taylor series expansion of the motion near the reference position $\phi =0$ in the form,

(4) \begin{equation} \mathbf{D}(\phi )= I + \mathbf{D}_1\phi + \mathbf{D}_2\frac{\phi ^2}{2}+\mathbf{D}_3\frac{\phi ^3}{6} + \ldots \quad, \end{equation}

where the matrices $\mathbf{D}_i$ are calculated to be,

(5) \begin{equation} \mathbf{D}_1= \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},\quad \mathbf{D}_2= \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & b_2 \\ 0 & 0 & 0 \end{bmatrix},\quad \mathbf{D}_3= \begin{bmatrix} 0 & 1 & a_3 \\ -1 & 0 & b_3 \\ 0 & 0 & 0 \end{bmatrix}. \end{equation}

This shows that every planar motion is defined by the set of constants $b_2$ , $a_3$ and $b_3$ to the third order, which are known as instantaneous invariants, Bottema and Roth [Reference Bottema and Roth32].

The trajectory $\mathbf{P}(\phi )$ in $F$ traced by a point $\mathbf{p}=(x,y)$ in the frame $M$ following the motion $\mathbf{D}(\phi )$ defined by (4) is given by,

(6) \begin{equation} \mathbf{P}(\phi )=\mathbf{p} + \mathbf{P}_1\phi + \mathbf{P}_2\frac{\phi ^2}{2} + \mathbf{P}_3\frac{\phi ^3}{6}+\ldots, \end{equation}

where

(7) \begin{equation} \mathbf{P}_1=\begin{Bmatrix} -y\\ x \end{Bmatrix},\quad \mathbf{P}_2=\begin{Bmatrix} -x\\ -y +b_2 \end{Bmatrix},\quad \mathbf{P}_3=\begin{Bmatrix} y +a_3\\ -x +b_3 \end{Bmatrix}. \end{equation}

This equation can be viewed as defining all of the trajectories $\mathbf{P}(\phi )$ associated with all points $\mathbf{p}=(x,y)$ in the moving plane $M$ .

We study the differential properties of all of the trajectories in the moving body in order to obtain the inflection circle, the Euler-Savary equation, and the cubic of stationary curvature.

3.1.3 Differential properties

The local properties of each trajectory in the moving body are studied using the unit tangent $\mathbf{T}$ and unit normal $\mathbf{N}$ vectors described in do Carmo [Reference do Carmo29] and Tapp [Reference Tapp30]. Using (6), we can compute these vectors to be,

(8) \begin{equation} \mathbf{T}= \frac{\mathbf{P}^\prime }{|\mathbf{P}^{\prime }|}=\frac{1}{(x^2+y^2)^{1/2}}\begin{Bmatrix}-y \\ x\end{Bmatrix},\quad \mbox{and}\quad \mathbf{N}=\mathbf{T}^\perp = \frac{1}{(x^2+y^2)^{1/2}}\begin{Bmatrix}-x \\ -y\end{Bmatrix}, \end{equation}

where $\mathbf{T}^\perp$ is $\mathbf{T}$ rotated $\pi/2$ counterclockwise. The prime in these equations is the traditional notation for derivatives with respect to the curve parameter, in our case $\phi$ .

The curvature $\kappa$ of a trajectory $\mathbf{P}(\phi )$ is calculated using the formula from differential geometry,

(9) \begin{equation} \kappa = \frac{\mathbf{P}^{\prime }\times \mathbf{P}^{\prime \prime }}{(\mathbf{P}^{\prime }\cdot \mathbf{P}^{\prime })^{3/2}} = \frac{x^2+y^2-b_2y}{(x^2+y^2)^{3/2}}. \end{equation}

This equation shows that the set of points $\mathbf{p}=(x, y)$ in the moving body with trajectories $\mathbf{P}(\phi )$ having zero curvature, that is with $\kappa =0$ at the instant $\phi =0$ , satisfy the equation,

(10) \begin{equation} \mathcal{I}\,{:}\,x^2+y^2-b_2y = 0, \end{equation}

which defines a circle, known as the inflection circle, Fig. 2.

Figure 2. The canonical coordinate system simplifies the equations that define tangent $\mathbf{T}_A$ and normal $\mathbf{N}_A$ vectors for a trajectory $\mathbf{A}$ , as well as for its the center of curvature $\mathbf{C}_A$ .

The center of curvature $\mathbf{C}_P$ for each trajectory $\mathbf{P}(\phi )$ is defined by the formula,

(11) \begin{equation} \mathbf{C}_P = \mathbf{p} + \frac{1}{\kappa }\mathbf{N}, \end{equation}

where $\rho =1/\kappa$ is the radius of curvature. The center of curvature defines a circle with radius $\rho$ that fits the trajectory to the second order. Substitute (8) and (9) into the above equation to obtain,

(12) \begin{equation} \mathbf{C}_P = \begin{Bmatrix} x \\ y \end{Bmatrix} + \frac{x^2+y^2}{x^2+y^2-b_2y}\begin{Bmatrix}-x \\ -y\end{Bmatrix}. \end{equation}

This definition of the center of curvature (12) takes a simpler form using polar coordinates in the canonical frame and yields the Euler-Savary equation, Uicker et al. [Reference Uicker, Pennock and Shigley33].

3.1.4 Euler-Savary equation

Consider a ray through a point $\mathbf{p}$ and the origin $\mathbf{V}$ of the canonical coordinate system, and introduce polar coordinates, so $\mathbf{p}=(R\cos \theta, R\sin \theta )$ . Recall that $\mathbf{V}$ is the instant center of rotation for the moving body, which means that the tangent vector $\mathbf{T}$ of the trajectory $\mathbf{P}(\phi )$ is perpendicular to this ray, and the normal vector $\mathbf{N}$ is along this ray. Using these polar coordinates, the curvature $\kappa$ is given by,

(13) \begin{equation} \kappa = \frac{R-b_2\sin \theta }{R^2}. \end{equation}

The center of curvature can be computed to be,

(14) \begin{equation} \mathbf{C}_P= R\begin{Bmatrix} \cos \theta \\ \sin \theta \end{Bmatrix} - \frac{R^2}{R-b_2\sin \theta }\begin{Bmatrix} \cos \theta \\ \sin \theta \end{Bmatrix}, \end{equation}

where it becomes clear that $\mathbf{C}_P$ lies on the ray through $\mathbf{p}$ at the angle $\theta$ and at the distance $R_C=R-\rho$ from the origin.

This ray intersects the inflection circle at the distance $R_I$ given by

(15) \begin{equation} R_I = b_2\sin \theta. \end{equation}

Substitute $R_I$ into the formula for the radius of curvature $\rho =R-R_C$ to obtain,

(16) \begin{equation} R-R_C=\frac{R^2}{R-R_I}\quad \mbox{or}\quad R^2=(R-R_C)(R-R_I), \end{equation}

which is known as the Euler-Savary equation, Uicker et al. [Reference Uicker, Pennock and Shigley33].

The Euler-Savary equation provides a formula for the location of the center of curvature $\mathbf{C}_A$ associated with the trajectory of a point $\mathbf{A}$ using the radial distances from the velocity pole to the point $\mathbf{A}$ and to the point of intersection $\mathbf{J}_A$ of this ray with the inflection circle. Repeat this calculation to obtain the center of curvature $\mathbf{C}_B$ for a second point $\mathbf{B}$ in order to obtain the four-bar linkage $\mathbf{C}_A\mathbf{ABC}_B$ that has a coupler motion that matches the moving plane to the second order, Fig. 3.

Figure 3. The Euler-Savary equation relates the radial distance to the center of curvature $R_{C,A}$ to the radial distance $R_A$ and the radial distance $R_{I,A}$ to the intersection with the inflection circle.

3.1.5 Ball’s point

We now consider the rate of change of the curvature of trajectories in the moving plane. Compute the derivative of the formula for curvature $\kappa$ in (9) to obtain,

(17) \begin{equation} \frac{d\kappa }{d\phi } = \frac{(\mathbf{P}^{\prime }\times \mathbf{P}^{\prime \prime \prime }) (\mathbf{P}^{\prime }\cdot \mathbf{P}^{\prime }) -3(\mathbf{P}^{\prime }\times \mathbf{P}^{\prime \prime })(\mathbf{P}^{\prime }\cdot \mathbf{P}^{\prime \prime })}{(\mathbf{P}^{\prime} \cdot \mathbf{P}^\prime )^{5/2}}. \end{equation}

Substitute (6) into this equation to obtain a formula for the rate of change of curvature in the reference position $\phi =0$ for every point $\mathbf{p}=(x,y)$ in the moving plane,

(18) \begin{equation} \kappa _1 = \frac{-(x^2+y^2)(a_3x+b_3y)-3b_2x(x^2+y^2-b_2y)}{(x^2+y^2)^{5/2}}. \end{equation}

The points with trajectories having a zero rate of change of curvature, that is with $\kappa _1=0$ , maintain the same curvature to the third order. These trajectories are said to have stationary curvature, and are defined by points that lie on the cubic curve,

(19) \begin{equation} \mathcal{C}: (x^2+y^2)(a_3x+b_3y)+3b_2x(x^2+y^2-b_2y)=0, \end{equation}

known as the cubic of stationary curvature.

The intersection of the cubic of stationary curvature and the inflection circle defines a point in the moving body that follows a straight line to the third order known as Ball’s point, Fig. 4.

Figure 4. Points on the cubic of stationary curvature follow circular trajectories to the third order, and Ball’s point follows a straight-line trajectory to the third order.

3.1.6 Differential kinematic synthesis

Kinematic synthesis of a four-bar linkage for movement of a body $M$ near a reference position seeks point trajectories $\mathbf{A}$ and $\mathbf{B}$ that lie on circles to the third order. These points lie on the cubic of stationary curvature and are used as the moving pivots of a four-bar linkage.

The centers of curvature $\mathbf{C}_A$ and $\mathbf{C}_B$ for these trajectories are found using the Euler-Savary equation, which become the fixed pivots for bars connecting the moving body $M$ to the ground frame $F$ . The result is a four-bar linkage $\mathbf{C}_A\mathbf{A}\mathbf{B}\mathbf{C}_B$ with a coupler line $\mathbf{AB}$ that follows the movement of $M$ to the third order.

Finally, we use Ball’s point to obtain a flat-sided coupler curve that we use for the foot trajectory of our leg mechanism. See Fig. 5.

Figure 5. For a choice of moving pivots $\mathbf{A}$ and $\mathbf{B}$ , the centers of curvature $\mathbf{C}_A$ and $\mathbf{C}_B$ form the fixed pivots. The resulting linkage $\mathbf{C}_A\mathbf{ABC}_B$ guides Ball’s point $\mathbf{P}$ along a flat-sided coupler curve.