MEC4127F: Introduction to Robotics

Chapter 2: Rigid-body motion

2.1 Introduction

2.2 Reference frames

2.2.1 The concept of reference frames

The concept of reference frames is very important in robotics, as it is the framework that is used to construct a robot’s pose (its position and orientation). For this reason, we will begin our robotic journey here. We will start by considering a robot with a single rigid body. The fact that this is a rigid-body robot means that the distance between any two points on the robotic platform will remain constant regardless of the robot’s position and orientation. As a result, we can define one point on the body as the origin of our now defined body-frame, labelled , and attach a 3D Cartesian coordinate system at this origin (see Figure 2.1).

Definition: Coordinate frame

Figure 2.1: Generic rigid-body platform (shown in blue) with body-frame coordinate system attached (green). World-frame coordinate frame, labelled , shown in red.

psi = 37*pi/180; %set rotation angle about z

 

R_b2w = [cos(psi) -sin(psi) 0; %rotation matrix

sin(psi) cos(psi) 0;

0 0 1]

R_b2w = 3×3

0.7986 -0.6018 0 0.6018 0.7986 0 0 0 1.0000

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([1 1 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B",MeshFilePath='multirotor.stl')

axis equal

view(2)

 

%set x-y plot limits

xlim([-0.63 1.93])

ylim([-0.47 2.09])

While robot frame may be a more intuitive descriptor, it is common practice to use the term body frame when referring to the frame that is attached to our robot body. With reference to Figure 2.1, this coordinate system consists of 3 orthogonal (perpendicular) vectors originating from the defined origin. We will label this reference frame (in this case the "B" corresponds to body-frame), with the first two reference frame axes following as and . Our third principle axis in , , points out of the page to simplify the initial discussion, and is by definition perpendicular to and using right-hand rule convention, namely . The "" symbol (which you can refer to as "hat", or "circumflex") is used to show that the principle axes are unit vectors – they have a vector length of one, namely .

We can similarly attach an additional coordinate system to the room that we are in. This will allow us to specify locations in a space that does not vary as the robot is displaced (moves around). The room is stationary and has no dynamics (ignoring the negligible effects of the earth rotating when considering a small local frame of interest) and we will call this our world-frame (or spatial-frame), with the corresponding label of. Where we choose the origin of our world-frame has no particular importance and comes down to convenience for the user. It makes no difference whether I choose the centre of the room or one of the corners as the origin — I can always adjust my understanding of the static space I am in by offsetting the position appropriately (more on this in the coming sections).

• a body-frame, , attached to our rigid-body platform; and
• a world-frame, , attached to the room.

Now if we want to describe the robot’s Cartesian position (in particular, described completely by the robot’s centre of mass), we just find the origin of with respect to . Note that as the robot translates (moves in a rectilinear motion), the robot position described in the world frame will change, but the position described in the body frame will remain the same (the centre of mass is defined to always be at the origin of our body-frame axis system). This is an important concept that relates to how the observer perceives information – standing stationery in the room, we observe the robot moving around, but from the robot’s perspective, we are the ones moving!

Similar to how we were able to describe our robot position in the world-frame, we can also describe our robot’s orientation in the world-frame, albeit with some additional effort. If we strategically place the robot in our room, such that the origins of and are coincident (existing at the same point in space), then the body-frame axes (specifically, where they point), described relative to the world-frame, give us all the information required to define the robot orientation. As the robot rotates in some arbitrary manner, the direction of the body-frame axes (represented in the world-frame) will change, and this will capture the changing orientation of the robot. This does not mean that we can only describe the rotation of the robot when the world-frame and body-frame origins are coincident. Because of our definition of the two coordinate systems, we can always assess the orientation of our robot by first mathematically shifting the position of the robot to our world-frame centre (again removing the effect of translation) and then evaluating the relationship between the two coordinate systems. In fact, as we will explain later, this process of shifting the origin is equivalent to introducing a third, intermediate, reference frame. That is, a reference frame that has the same orientation of , but with an origin that is coincident with frame .

Example: Remote control car

This is illustrated using the code block below, where you could image the axis projecting out of our eyes, and pointing in the forward direction of the remote control car.

psi = 23*pi/180; %set rotation angle about z

 

R_a2b = [cos(psi) -sin(psi) 0;

sin(psi) cos(psi) 0;

0 0 1]; %rotation matrix

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([2 0 0],rotm2quat(R_a2b),"FrameAxisLabels","on","FrameLabel","B",MeshFilePath='groundvehicle.stl',FrameSize=0.3)

axis equal

view(2)

 

%set x-y plot limits

xlim([-0.63 2.18])

ylim([-0.47 2.09])

2.2.2 Stationary reference frames

It is important to note that all frames considered in this course are stationary, inertial, frames. When referring to a frame, such as , we mean a motionless frame that is instantaneously coincident with a frame that is fixed to a (possibly moving) body. This is important to keep in mind, as you may have had a dynamics course that used non-inertial moving frames attached to rotating bodies.

2.2.3 Representing vectors in different frames

The previous section gave a mostly qualitative overview of how the body and world frames are constructed to facilitate determining the position and orientation of a rigid-body robot. Because we are working in 3D space, our rigid-body robot’s position can be described by a vector. Following from our discussion in Section 2.2, a vector has no meaning (it is ambiguous) until we identify which reference frame it is defined in. In fact, the description of a vector collapses to a point with no indication of its origin (there is no associated direction or magnitude).

In general, we will denote the reference frame of the vector by using a leading superscript of X, which indicates that the vector is described in coordinate system . Looking at Figure 2.3, given an arbitrary point , we can represent the corresponding vector in the body frame as , and in the world frame. Note that depending on the reference frame that the vector is attached to, the length and direction of said vector will, in general, be different.

Figure 2.3: The point can be described as a vector once it is defined relative to a particular reference frame (the vector originates from the frame origin). is the vector with respect to , and is the vector with respect to .

2.2.4 Translating reference frames

Imagine now that we are interested in finding the location of the body-frame (frame ) origin, with respect to the world frame (frame ). We will call this vector , which depicts the translation vector from the frame origin to the origin of frame — the offset between the two origins (see Figure 2.4),

Figure 2.4: Description of translation vector and component breakdown.

It should come as no surprise that the corresponding vector in is . This holds true regardless of how the robot moves. is written explicitly as

where the lowercase subscripts refer to the particular component of . Similar to how we previously defined our body-frame axes, we can express our world-frame coordinate system as

• — our world-frame x-axis,
• — our world-frame y-axis,
• — our world-frame z-axis.

In other words, (for example) is the component of that is parallel to (see Figure 2.4), where by definition. Note that is a linear combination of the three basis vectors that span the world-frame coordinate system.

The code block gives an illustration of how adjusting moves the origin of away from .

the = 66*pi/180; %adjust orientation

 

tx =3; %adjust x component of translation vector

ty =1.1; %adjust y component of translation vector

 

t_b2w = [tx ty 0]; %full translation vector

 

R_b2w = [cos(the) -sin(the) 0; %principal z-axis rotation matrix

sin(the) cos(the) 0;

0 0 1];

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms(t_b2w,rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

 

title(['${^W{\bf t}_B}=[$',num2str(t_b2w,2),'$]^T$'],'Interpreter','latex') %add text to show rotation angle on figure

xlabel('x axis'),ylabel('y axis')

 

view(2)

axis equal %make x- and y-axes proportionate

xlim([0 5]),ylim([0,5])

 

2.3 Rotation matrices

2.3.1 Derivation

As alluded to in the previous section, if we can describe the set of body-frame axes, {, , } in the world frame, then we have a description of our orientation. In other words, we want to find a mapping that relates the body-frame axes to the world-frame, namely . Considering only for now, and assuming the origins of our two reference frames are coincident (for now), we want to find the corresponding , which would represent the projection of onto the three world-frame axes. We could have also written as , but the addition of the leading superscript is redundant and is omitted for sake of clarity. The projection of into can be found by using the dot product (also known as the inner product), where the dot product is generically defined as , where and are arbitrary vectors, and α is the angle between the two vectors. When dealing exclusively with unit vectors, the dot product simplifies to .

Figure 2.5: A geometric projection of onto the principal axes of using the unit vector dot product.

With reference to Figure 2.5, we can now find using the unit vector dot product, which yields

We have essentially broken up the body-frame vector into the components parallel to the three world-frame principal axes. We can repeat this on the other two body-frame axes, and if we subsequently stack them all together into a 3×3 matrix, we obtain a special matrix known as a rotation matrix (specifically referred to as a direction cosine matrix in this configuration based on the use of nine cosine operations):

The superscript and subscript in indicates that the rotation matrix is describing the orientation of with respect to . By extension, maps (or rotates) vectors from to . For example, we can find using this newly defined relationship of

which is analogous to our calculation further above. We now have a formal description of a matrix mapping that can take in any arbitrary vector described in (e.g. ), and rotate it into the world-frame (e.g. ) using :

Looking at the right-hand side of the above equation, we see that the subscript of "cancelled out" the leading superscript of (as they were both B) and replaced it with a leading superscript W. This is a nice way of sanity checking the rotation operations.

Following this, we can pre-multiply both sides of by to give us

Based on this result, describes the mapping from the world frame into the bodyframe, , and as such, we can express this new mapping as (note the subscript and superscript swap when there is the transpose operation):

Again, the subscript-superscript "cancellation" of Won the right-hand side of the above equation infers that the mapping makes ordered sense. So we now have a method that can be used to map arbitrary vectors back and forth between any two reference frames (assuming their origins are coincident), and that mapping (our rotation matrix) inherently describes the orientation of the one frame relative to the other frame. In particular, will describe the orientation of the body-frame (which by definition is also the orientation of our robot) relative to the world-frame, and will describe the orientation of the world frame relative to the body frame. It is important to note that rotation matrices encode orientation - we would still need to extract the angular information in some or other form (more on this in Chapter 3).

Note that if the two reference frames were perfectly aligned, our rotation matrix would simply be , where

is the 3×3 identity matrix. This would imply , , and (all three axes sitting on top of each other, respectively). The rotation matrix, , is actually no ordinary 3×3 matrix. Because each column of the rotation matrix comprise unit vectors (rotating a unit vector preserves its length) that are perpendicular to each other, this means that is an orthogonal matrix.

Definition: Orthogonal matrix

and have a determinant of . By extension, the equation above implies that the transpose of the rotation matrix is equal to its inverse, . Note that by explicitly evaluating the equation above using the parameterisation

we can see that implicitly enforces the unit-norm condition ( ) of

as well as the orthogonality condition () of

syms r11 r12 r13 r21 r22 r23 r31 r32 r33 real %symbolic definition of variables, all defined to be in the set of reals

R = [r11 r12 r13; r21 r22 r23; r31 r32 r33]; %rotation matrix, parameterised by symbolic variables above.

R'*R==eye(3) %showing requirement for orthogonal matrix.

ans = 

2.3.2 Special orthogonal group

The definition of the rotation matrix above applies to both left-handed and right-handed reference frames. As mentioned in Chapter 1, we will only consider right-handed rotation frames and rotations, which has the implicit requirement of (the left-handed counterpart would be ) when considering the rotation matrix form of

It can be shown that the determinant of a right-handed rotation matrix will always have a value of , namely

.

The set of rotation matrices that have a determinant of forms the special orthogonal group, abbreviated as SO(3).

Definition: Special orthogonal group SO(3)

The special orthogonal group SO(3), also known as the group of rotation matrices, is the set of all real matrices that satisfy (i) and (ii) .

Example: Special orthogonal group

We first check to see that is orthogonal, using which is easily shown to be true. We then either calculate the determinant of , or check whether the cross product of the first two columns is equal to the third column, based on being a necessary condition for SO(3). This leads to

which informs us that is left-handed, and therefore not part of the special orthogonal group SO(3).

2.3.3 Rotation in a plane

In the special case that all motion occurs in a plane (2D surface), such as a wheeled robot traversing on a flat road, we can instead consider position as a 2D vector, along with the rotation matrix as being of size .

the = 25*pi/180; %rotation angle about z

 

R_b2w = [cos(the) -sin(the) 0; %principal z-axis rotation matrix

sin(the) cos(the) 0;

0 0 1];

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([0 0 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

 

text(0.1,0.02,['\theta=',num2str(the,2)]) %add text to show rotation angle on figure

 

view(2) %view from perspective of z-axis

axis equal %make x- and y-axes proportionate

xlim([-1 1]),ylim([-1,1])

Given that θ is the angle between and , we can determine the mapping of into by

Similarly, can be described in as

Recalling that and omitting the third column, we arrive at the rotation matrix of

Definition: Special orthogonal group SO(2)

The special orthogonal group SO(2) is the set of all real matrices that satisfy (i) and (ii) .

2.3.4 Uses of rotation matrices

2.3.4.1 Representing an (encoded) orientation

The code block below generates three different reference frames: , , and , which all have differing orientations.

R_a2w = [0 -1 0; %rotation matrix describing orientation of {A} wrt {W}

1 0 0;

0 0 1];

R_b2w = [0 -1 0; %rotation matrix describing orientation of {B} wrt {W}

0 0 -1;

1 0 0];

 

%plot figures

figure,hold on

subplot(1,3,1), hold on, plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

 

subplot(1,3,2), hold on, plotTransforms([0 0 0],rotm2quat(R_a2w),"FrameAxisLabels","on","FrameLabel","A")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

 

subplot(1,3,3), hold on, plotTransforms([0 0 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

With reference to the code block above and the resulting figure, the (encoded) orientation of with respect to is given by

1. the x-axis of points in the direction of ,
2. the y-axis of points in the direction of ,
3. the z-axis of points in the direction of .

The orientation of is therefore encoded within the above rotation matrix. We will discuss "decoding" the orientation in detail in Chapter 3. Based on our orthogonality property of we also know that If represents the orientation of with respect to , then the inverse of this matrix (which we have shown is equal to the transpose), represents the orientation of with respect to , namely

This can be generalised to any set of reference frames. We can also find the relative orientation of with respect to . From the code block above we are given

To find , which represents the orientation of with respect to , we pre-multiply the equation above with to get

Note the "cancellation" of the W subscript and superscript. This calculation can be easily checked numerically using the code block below to assess that each axis of is pointing in the correct direction — this can also be confirmed by the visualisation of and .

2.3.4.2 Representing a vector in a different frame

The rotation matrix allows us to represent vectors in any frame of interest. Consider the figure generated from the code block below. A common point, , is plotted in all three axes.

p = [2 0 0]'; %vector p wrt {W}

 

R_a2w = [0 -1 0; %rotation matrix describing orientation of {A} wrt {W}

1 0 0;

0 0 1];

R_b2w = [0 -1 0;

0 0 -1;

1 0 0]; %rotation matrix describing orientation of {B} wrt {W}

 

%plot figures

figure,hold on

subplot(1,3,1), hold on, plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plot3(p(1),p(2),p(3),'or',MarkerSize=5,MarkerFaceColor='red')

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

subplot(1,3,2), hold on, plotTransforms([0 0 0],rotm2quat(R_a2w),"FrameAxisLabels","on","FrameLabel","A")

plot3(p(1),p(2),p(3),'or',MarkerSize=5,MarkerFaceColor='red')

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

subplot(1,3,3), hold on, plotTransforms([0 0 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

plot3(p(1),p(2),p(3),'or',MarkerSize=5,MarkerFaceColor='red')

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

This point has no magnitude or direction associated with it until we assign it to a particular reference frame. Starting first with frame , we define the associated vector, . This description should make sense to you in the left-most figure. To describe this point in , we need to change the reference frame representation, which can also be thought of as rotating the vector accordingly. This is accomplished using

.

.

This result can be verified by consulting the centre figure showing the point with respect to . We can follow the same procedure to determine but that will be left to you as homework!

If we already know and and instead want to find , then we can pre-multiply both sides of by , which yields

.

2.3.4.3 Rotating a vector or frame

Specifically, if we rotate about with an angle of , we obtain . Similarly, is obtained by rotating about with an angle of . This is related to principal axis rotations, which we will discuss in detail in Chapter 3. Note that if a vector is fixed in a frame, for example a point of interest on a rigid-body robot (not the case in this example), then any subsequent rotation of that frame would also rotate the vector. This will be discussed in more detail in the latter chapters.

R_w2w = [1 0 0; %rotation matrix describing orientation of {W} wrt {W}

0 1 0;

0 0 1];

 

R_z90 = [cos(pi/2) -sin(pi/2) 0; %rotation matrix describing rotation of pi/2 about z axis

sin(pi/2) cos(pi/2) 0;

0 0 1];

 

R_a2w = R_w2w*R_z90;

 

R_y90 = [cos(-pi/2) 0 sin(-pi/2); %rotation matrix describing rotation of -pi/2 about y axis

0 1 0;

-sin(-pi/2) 0 cos(-pi/2)];

 

R_b2w = R_a2w*R_y90;

 

%plot figures

figure,hold on

subplot(1,3,1), hold on, plotTransforms([0 0 0],rotm2quat(R_w2w),"FrameAxisLabels","on","FrameLabel","W")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

subplot(1,3,2), hold on, plotTransforms([0 0 0],rotm2quat(R_a2w),"FrameAxisLabels","on","FrameLabel","A")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

subplot(1,3,3), hold on, plotTransforms([0 0 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

set(gca,'XColor', 'none','YColor','none', 'ZColor','none'), axis equal

2.4 Generalised vector mapping

In general, two reference frames will be offset from each other by some amount, as was shown in Section 2.2.4. We (should) know that we can add any two vectors together only if they are in the same reference frame - we are essentially doing this when we do any kind of rudimentary addition. With reference to Figure 2.7, if describes the location of the frame origin in , and given an arbitrary body-frame vector of , we can find the corresponding vector in the world frame using

In other words, we have first rotated such that points in the right direction, followed by offsetting (or translating) by . The equation above describes the generalised vector mapping from the body frame to the world frame, which is abstractable to any two reference frames. Note that when , the equation above reduces to (i.e. when and have coincident origins).

So in general, a rotation and translation is required to fully describe the pose of one frame relative to another. In fact, as previously mentioned, we have implicitly introduced a third intermediate frame, , which has the same origin as that of and possesses the same orientation as frame with respect to frame . An illustration of three reference frames are shown below.

a = 45*pi/180; %rotation angle

t_b2w = [1 1 0]'; %location of frame {B} origin wrt {W}

 

R_b2w = [cos(a) -sin(a) 0; %rotation matrix describing rotation of a about z axis

sin(a) cos(a) 0;

0 0 1];

R_a2w = R_b2w;

 

%plot figures

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([0 0 0],rotm2quat(R_a2w),"FrameAxisLabels","on","FrameLabel","A")

plotTransforms(t_b2w',rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

 

view(2)

axis equal

This was what we previously used to describe the rotation matrix when ignoring the displacement between two frames. Note that is constructed by offsetting by .

If we want to repose equation such that is the subject of the formula (in other words, find the mapping from to ), all we need to do is pre-multiply by the rotation matrix inverse (equivalent to its transpose) and then rearrange,

where . In other words, the mapping of requires application of the rotation matrix (which defines the vector with respect to the correct reference frame), and then multiplying the resulting vector by (making the vector point in the opposite direction).

2.5 Homogeneous transformation matrices

2.5.1 Derivation

We begin by representing vectors and in homogeneous coordinates of the form

.

Note that all we have seemingly done is appended a "1" to the two position vectors (there is more relevance to these coordinates, especially in computer vision applications, but we will not explore it this course). We then determine the matrix mapping from to , which is based on refactoring and is given by

Note the last row is always balanced ( ), and does not affect the first three rows. This matrix mapping,

is known as the homogeneous transformation matrix and is a compact method of performing vector mappings. Note that this transformation matrix is a function of the rotation matrix, , and translation vector, , and as such encodes the full pose of frame in .

Definition: Special Euclidean Group SE(3)

The special Euclidean group SE(3), also known as the group of rigid-body motions or homogeneous transformation matrices in , is the set of all real matrices of the form

where SO(3), , and is a column vector.

2.5.2 Inverse transform

It can be shown that the inverse of our homogeneous transform, , can be constructed by inverting the rotation and translation parts independently, which yields

2.5.3 Sequential matrix transforms

We have now established a convenient, consistent notation to convert vectors from one coordinate system to another. Remember that will encode the robot's relative pose between two reference frames.

If we were to introduce an arbitrary additional frame, , and we wanted to describe the pose of frame relative to frame , we would similarly use . This allows us to map vectors from from frame to frame . For example, we can map through this transform using

we can rewrite as

By extension, it follows that (note the inner sub- and superscripts "cancelling"). This means that we can chain together homogeneous transforms when describing relative poses in the same way that we can chain together rotation matrices when describing orientation. As an example, our previous process of rotating the vector, followed by offsetting it, as shown in Section 2.4, is actually equivalent to the product of two homogeneous transformations — one describing pure rotation, and one describing translation.

For the case when reference frame and are different only by orientation, the corresponding transformation matrix is given by

where represents the relative orientation. For the case of a pure translation between frame and , the corresponding transformation matrix is given by

where is a identity matrix representing the null rotation, and translates the origin of to .

Multiplying the two transforms together (noting the rightward multiplication order as we move from ), we arrive at

This implies that , and .

The ability to sequentially move between different frames is useful in robotics, as we will see in future Chapters. It must be stressed that the matrix multiplication order matters: . This error is fortunately easy enough to avoid if we use the superscript-subscript notation as shown in these notes.

2.5.4 Motion in a plane

The previous section presented the generalised form of the transformation matrix. In the case of pure rotation in a plane, as shown previously, the orientation could be represented by a rotation matrix in SO(2) — namely a rotation matrix of the form

where θ is the angle of rotation of a particular frame. By extension, and given that a planar position is suitably represented by a 2-vector, , the corresponding homogeneous transformation matrix is given by

Definition: Special Euclidean Group SE(2)

The special Euclidean group SE(2) is the set of all real matrices of the form

where SO(2) and is a column vector.

which is evaluated using the atan2 function. This is done so that is determined within its full range.

2.5.5 Uses of transformation matrices

2.5.5.1 Representing a configuration

The code block below generates three frames — , , and — all with different positions and orientations.

R_b2w = [0 0 1; %rotation matrix describing orientation of {B} wrt {W}

0 -1 0;

1 0 0];

 

R_c2w = [-1 0 0; %rotation matrix describing orientation of {C} wrt {W}

0 0 1;

0 1 0];

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([0 -2 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

plotTransforms([-1 1 0],rotm2quat(R_c2w),"FrameAxisLabels","on","FrameLabel","C")

 

view(2)

axis equal

With reference to the code block above and the resulting figure, the rotation matrices of and with respect to are given by

The translation vectors of and with respect to are given by

.

We can combine this information to represent the corresponding transformation matrices describing the configuration (or pose) of and with respect to , namely

We can use this matrix formulation to determine the pose of with respect to , for example, based on our knowledge that

,

The calculation is verified using the code block below. The frame origin and orientation in can also be ratified by consulting the figure.

2.5.5.2 Representing a vector in a different frame

The transformation matrix allows us to represent vectors in any frame of interest. Consider the figure generated from the code block below, where point is included in the previous figure.

R_b2w = [0 0 1; %rotation matrix describing orientation of {B} wrt {W}

0 -1 0;

1 0 0];

 

R_c2w = [-1 0 0; %rotation matrix describing orientation of {C} wrt {W}

0 0 1;

0 1 0];

 

p = [-1 -1 0]'; %vector p described wrt {W}

 

%plot figure

figure,hold on

plotTransforms([0 0 0],[1 0 0 0],"FrameAxisLabels","on","FrameLabel","W")

plotTransforms([0 -2 0],rotm2quat(R_b2w),"FrameAxisLabels","on","FrameLabel","B")

plotTransforms([-1 1 0],rotm2quat(R_c2w),"FrameAxisLabels","on","FrameLabel","C")

plot3(p(1),p(2),p(3),'ko',MarkerFaceColor='k')

 

view(2)

axis equal

This point has no magnitude or direction associated with it until we assign it to a particular reference frame. Starting first with frame , we define the associated vector, . To describe this point in , we need to change the reference frame respresentation, which is accomplished using our transformation matrix,

.

The same process can be followed to determine but will be left to you as an exercise.

2.5.5.3 Rotating a vector or frame