Abstract
The machinability of a surface describes its ability to be machined and the factors which affect this. These are independent of any material properties or cutting parameters but instead reflect an ability to replicate a desired tool path motion with sufficient control of the material removal process. Without this control there is a potential for surface defects and costly finishing stages.
Five-axis CNC milling machines are commonly used for machining complex free-form shapes. The processes required to obtain CNC instructions for a machine tool, starting from a target surface, are presented. An overview is first given and later formalised with mathematical methods. Specifically, a moving cutting tool is characterised by a tool path motion. Interpreting the moving cutter in terms of moving machine axes provides a diagnostic tool for detecting machining errors.
Examination of two case studies reveals different types of errors, machine-dependent and machine-independent. The contribution of geometry to machine-independent errors is discussed and related back to the machinability of a surface.
1 Introduction
Computer Aided Design and Manufacture (CAD/CAM) provides a highly-automated process for the machining of components. The term machining is used to describe a process that begins with some raw material which is gradually removed via cutting until a desired shape is achieved. The machinability of a part thus refers to its ability to be machined and the factors which affect this [1]. Five-axis CNC milling machines are commonly used for machining parts with complex free-form shape due to their ability to control the position and orientation of a cutter.
The manufacturing process begins with the design of a part geometry inside a CAD environment [13]. The end goal is to machine this part geometry as efficiently as possible and for it to be of a sufficiently high quality. An immediate measure of quality is the dimensional accuracy of the part to that of the CAD model. Another consideration, which is decisive in whether or not a part is deemed acceptable, lies in its visual appearance. Any aesthetic irregularities on the part surface require costly finishing stages even if the part is within dimensional tolerance.
An important factor in machining a satisfactory part is sufficient control of the cutting tool and thus the cutting conditions. This control is governed by the Computer Numerical Control (CNC) unit of the machine tool. The CNC controller does not exactly replicate the desired motion due to physical restraints. The difference between desired and actual motion affects the cutting conditions and can therefore be used to predict surface defects. Since the tool path motion is defined from the surface geometry the effect of geometry in causing these surface defects is important to understand. The machinability of a surface is used here to describe its ability to be machined independent of any material properties. In particular it is used here to describe the properties of a tool path motion generated from the surface and the ability to replicate this motion with sufficient control of the material removal process.
The CNC controller is able to manipulate the position and orientation of the cutter with respect to the workpiece by moving the machine axes. These movements must obey physical constraints in reality, such as limited speed and acceleration, in accordance with the laws of physics. Constraints on axes speed and acceleration are hard-coded into the CNC controller. Thus when a motion does not abide by these constraints an alternative to the desired motion is experienced. Positional tolerance of the cutter is of high priority in the CNC controller and the most likely effect is a slow down in cutting feed. This affects the surface integrity of the part, possibly producing dwell marks (Fig. 7). Since these affect the visual appearance of the part, analysis of the machine axes kinematics can be used to predict the occurrence of machining flaws.
In this paper two simplified case studies are examined which are based on real life examples where surface defects have occurred. Analysing irregularities in the motion of a cutting tool provides an explanation for the occurrence of these defects. Furthermore, a relationship between part geometry and potential sources of defects is established via analysis of the tool path motion.
The structure of the paper is described as follows. Section 2 presents an overview of the machining process with regards to forming tool paths from a part geometry [2]. Section 3 formalises these processes with the relevant mathematical preliminaries. Cutter poses are defined with the introduction of the workpiece coordinate system. The connection to machine axes is presented through the kinematic chain of the machine tool leading to the machine coordinate system. A relationship between these coordinate systems for a tool path motion is then presented via the Jacobian matrix.
Section 4 presents the first case study. A tool path is constructed to demonstrate machining errors identified as a feature of the machining singularity. The singularity is shown to correspond to a degenerate Jacobian matrix. Section 5 presents the second case study. The example considers a single tool path used for machining a simplified turbine blade. The machining errors here are shown to be linked to the geometry of the tool path motion. Section 6 discusses the link between the geometry and the machine axes’ behaviour. The paper finishes with some concluding remarks on the two different types of machining errors.
2 Background to the Machining Process and Tool Path Motion Analysis
This section presents an overview of the procedures involved with the machining process, beginning from the CAD geometry and finishing with movements of the machine axes. It is summarised by the flow diagram given in Fig. 1. The mathematical formalisation of these procedures is presented in Sect. 3.
In order to transform raw material (workpiece) into a desired shape, a machining strategy has to be developed. The first stage involves removing the bulk of the excess material in what is known as the roughing stage. The final machining stage traces over the surface geometry of the part removing the excess material remaining.
This finishing stage can be achieved in a variety of ways but all abide by the same principle. Given a target geometry to machine, a cutting tool is traced over the surface forming a sequence of connected tool paths. The envelope of the moving cutting tool must not intersect with the part geometry else over cutting (gouging) occurs. Furthermore, the volume represented by the envelope must remove all excess material leaving only the material coinciding with the part geometry. Thus the volumetric intersection of the tool path envelope with the CAD model should theoretically be the boundary surfaces of the part. Note that in practice the target surface geometry does not exactly coincide with the tool path envelope but is instead designed to be within a strict dimensional tolerance of the target shape. The formation of these tool paths is usually computed with the aid of Computer Aided Manufacturing (CAM) software [3].
The position and orientation of the cutter is referenced with respect to a coordinate system fixed relative to the workpiece. The combined position and orientation information is here referred to as a pose. When the cutter removes material in the finishing stage it touches the target surface at a particular point on the cutting tool edge. This point is referred to as the cutter contact (CC) point. However, the position of the cutter must be defined by a fixed point on the cutter called the cutter location (CL) point. This is usually defined at the tip of the cutter. Note that the location of the CC is not generally at a fixed displacement from the CL. Thus CL data is derived from varying offsets of the target surface which depend upon the orientation and geometry of the cutting tool (see Fig. 2). This offsetting can change the geometry of the CL data from that derived from the surface. However efforts are usually made to preserve the angle of the cutter to the surface normal and tangent direction in order to maintain a consistent chip pattern on the surface [4]. This geometric property proves insightful when considering the derivatives of the tool path motion.
The next stage in the manufacturing process involves converting the tool paths, in the form of positional CL and orientation data, into machine instructions for a CNC controller. These instructions, commonly known as NC code, depend upon what type of CNC machine tool is to be used. Information regarding the machine tool structure is required to determine the effect each axis has on the pose of the tool with respect to the workpiece. This information is contained within the kinematic chain which connects frames of reference across each axis of the machine as well as the frames of reference connecting the workpiece and cutter (more detail is given to the kinematic chain in Sect. 3). The kinematic chain can thus be used to define a locally one-to-one function between the tool poses of the CAM model and the five machine axis values that replicate each pose.
The tool path from the CAM software is discretized into a sequence of tool poses which are then converted into five corresponding machine axis values in the post-processing stage. This list of machine configurations forms part of the CNC instructions for the machine tool along with other machining parameters such as spindle speed. The CNC machine can then move the machine axes to interpolate each sequential configuration, machining the desired geometry (within tolerance). The amount of time desired between each pose is also stored in the NC code in an attempt to control the cutting speed (or feed rate) and thus cutting conditions. Alternatively an explicit feed rate can be given.
The cutting speed represents the relative speed between the cutter and the surface of the part. It is usually constant for each individual tool path to preserve cutting conditions. Consider the collection of CC points for such a tool path in the finishing stage. These derive from a curve on the target surface and are parameterised by some constant factor of the arc-length parameterisation for this curve. This provides enough information to fully define a tool path motion. It is a function that outputs the pose of the cutter for a given input of time. This concept is formalised in more detail in Sect. 3.
Given a tool path motion it is possible to convert the poses into machine axes values and analyse the changes in them with respect to time. These represent kinematical characteristics of the machine and therefore must abide by the laws of physics. For example, the acceleration for each axis is bounded by the amount of force (or power) that can be transmitted in accordance with Newton’s second law. The maximum speed must also be bounded not least for safety but also for the sustainability of the machine. In order to smooth the movement of a machine these characteristics are controlled by the CNC controller. The exact details of the control algorithms used are not in the public domain as the intellectual property belongs to the manufacturer. However, most abide by similar principles. Data corresponding to position, velocity and acceleration is measured at an instance. Decisions regarding how much power to deliver to the axes are then made by the controller from this data in a closed feedback loop [5]. A fundamental component of this decision making is limiting the maximum speed and acceleration of the machine axes. These bounds are hard-coded inside the controller. If the desired tool path motion exceeds these bounds then the controller must make adjustments. Even if the controller is able to maintain positional accuracy, the cutting speed cannot be maintained. These dwells cause irregular chip patterns and rubbing of the cutting tool. This in turn affects the surface finish of the part and may cause it to be rejected (Fig. 7).
Therefore, given a tool path, it is possible to detect these types of machining issues (here referred to as machining errors). The process involves generating the machine axes values at each instance of time and checking that they lie within predetermined bounds. Analysis of two case studies leads to distinct causes of machining errors. In the first case the excessive speeds and accelerations can be eliminated by changing the kinematic chain. This is akin to having a different machine tool perform the same tool path motion. For this reason they are classified as machine-dependent errors. In the second case however it can be shown that no kinematic chain can resolve the issue and the cause of the error lies in the geometry of the tool path motion. These are referred to as machine-independent errors. Since the tool path motion is defined from the surface of the part, the link with surface geometry and machining errors is discussed in Sect. 6.
The next section presents the mathematical processes required to form the tool path motion. Then using knowledge of the CNC machine tool structure, in the form of a kinematic chain, the relationship between tool poses and machine axes values is given. These form two separate coordinate systems. The Jacobian between these coordinate systems is then presented which proves useful for analysing how the coordinate systems change with respect to each other.
3 Mathematical Preliminaries
3.1 Coordinate Systems and the Kinematic Chain
A workpiece coordinate system is defined with respect to some fixed coordinate frame rigidly attached to the workpiece. The position of the cutter is described with a triplet of \(\mathbf{{V}}_W=(x,y,z)\) values. More precisely this is the CL point of the cutter which is normally chosen to be the tip of the cutter. The cutting tool rotates in the spindle about a line. The direction of this line is described with a unit vector, \(\mathbf{{O}}_W=(i,j,k)\), which represents the orientation of the cutter. The combined position and orientation coordinates, \(\mathbf{{P}}_W=(x,y,z,i,j,k)\) form the pose of the cutter. In the post-processing stage, the pose of the cutter must be converted into machine coordinates, \(\mathbf{{P}}_M=(M_1,M_2,M_3,M_4,M_5)\), representing the values of each machine axis. This requires information about the CNC machine tool regarding the arrangement of the axes relative to the workpiece and cutter.
A kinematic diagram is a graph that represents the connectivity of links and joints in a machine (see Fig. 3). Nodes of the graph represent parts of the machine and the edges between them represent joints. Each joint can be represented as a rigid-body transform between coordinate systems fixed at each link. A joint is considered an actuator if its movements can be controlled by the machine. The kinematic graph of a five-axis CNC machine therefore comprises five actuator joints connected to the workpiece and the cutting tool. Only serially connected machines are considered here. This implies that each actuator is connected in series and the kinematic diagram is a tree with world space as the root. The five actuators combined with the workpiece and cutter form a closed kinematic loop [6]. That is to say if one begins at a node and applies the rigid-body transforms of each sequential edge as one loops round the graph one arrives back at the original node with the same frame of reference. The machine body is fixed to the ground (world space) and remains stationary whilst the machine is moving. This is signified with a connection to ground and helps to define a fixed reference frame for each set of links and joints.
The pose of the cutter with respect to the workpiece is represented in the kinematic diagram by the edge connecting them. Thus given values of each actuator the pose of the cutter can be determined by looping round the kinematic chain from the workpiece to the cutter thorough the actuator joints. Since each edge represents a rigid-body transform, applying these transformations sequentially results in the rigid-body transform corresponding to the pose of the cutter.
Take the Hermle C600U machine tool for example. A schematic of the machine tool is given in Fig. 4. This machine consists of 3 translational axes (X, Y, Z) controlling the spindle position and two rotary axes (A, C) controlling the orientation of the workpiece mounted on the machine bed. For simplicity a reference (or world space) coordinate frame is chosen such that the origin corresponds to the intersection of the axes of rotation for the A and C rotary axes. The (x, y, z) directions align with the (X, Y, Z) translational movements of the spindle. The origin of the cutter’s reference frame is chosen at the tip of the cutter. The orientation of the cutter’s frame is aligned to the reference frame. The coordinate frame for the workpiece is chosen so that when the A and C rotary axes are set to zero it coincides with the reference frame. The kinematic diagram is given in Fig. 3.
The pose of the cutter can thus be inferred from the kinematic chain. Note the rigid-body transform from the reference frame to the cutting tool is simply a translation of (X, Y, Z). The rigid-body transform from the reference frame to the workpiece is a rotation of angle C in the Z direction followed by a rotation of angle A in the X direction. Thus the CL location is given by
Here A and C are the angles of the rotary axes and \(\mathbf{{R}}_i(j)\) is the rotation matrix about the i axis of angle j. Note that the subscript of W or M represents the coordinate frames chosen to determine the CL position, where M is the machine coordinate frame and W is the workpiece coordinate frame. The orientation of the cutter can also be determined by reorienting through the kinematic chain, starting with it aligned in the Z direction.
Given a cutter pose, \(\mathbf{{P}}_W=(x,y,z,i,j,k)\), the corresponding machine axes values can be calculated. The first task is to find the A and C values that produce the orientation \(\mathbf{{O}}_W=(i,j,k)\). This can be found by solving Eq. (2). Table 1 shows possible solutions [7]. The coordinates (X, Y, Z) can then be found from Eq. (1) by premultiplying by the appropriate rotation matrices.
Either the workpiece coordinates or machine axes values (machine coordinates) can be used to describe a pose. The workpiece coordinates characterise the geometry of the tool path whereas the machine coordinates characterise the machine behaviour. A useful tool for describing the link between coordinate systems is the Jacobian matrix.
3.2 The Jacobian Matrix
The change in machine coordinates over time represents the kinematic properties of the individual axes. If the speed or acceleration of these axes becomes too large then surface defects can occur. The change in workpiece coordinates is described with the tool path motion. Thus, in order to gain a better insight into how the moving cutter corresponds to movements in machine axes, derivatives of the coordinates should be considered.
The velocity of machine axes movements can be inferred from the velocity of the pose via the chain rule.
The \(\mathbf{{J}}\) term, referred to as the Jacobian matrix, describes the relationship between velocities in the two coordinates systems as a matrix transformation. The matrix is non-square because in workpiece coordinates the unit normal \(\mathbf{{O}}_W=(i,j,k)\) has only two degrees of freedom. Nonetheless, an upper bound on this matrix transformation can be obtained from analysis of the spectral norm defined as:
This equals the square root of the largest eigenvalue of \(\mathbf{{J}}^T\mathbf{{J}}\) [8] and is referred to here as the bound of the Jacobian.
3.3 Machine Singularities
The Jacobian, \(\mathbf{{J}}\), represents the change in machine coordinates with respect to workpiece coordinates. A similar matrix, \(\mathbf{{K}}\), for the change in workpiece coordinates with respect to machine coordinates can be derived as:
It can be calculated by differentiating the rigid-body transformation between cutter and workpiece as described in the kinematic chain.
If the rank of the matrix \(\mathbf{{K}}\) is less than 5 then some machine axes movements become redundant in that they do not affect the pose of the cutter relative to the workpiece. This occurs when \(\det (\mathbf{{K}}^T\mathbf{{K}})=0\) with the redundant movements corresponding to the eigenvector with zero eigenvalue. Furthermore the Jacobian, \(\mathbf{{J}}\), is undefined here. In this scenario there are not enough degrees of freedom in the system to accommodate all possible pose changes, this is identified as a machining singularity [9].
As \(\det (\mathbf{{K}}^T\mathbf{{K}})\rightarrow 0\) the spectral norm of the Jacobian \(||\mathbf{{J}}||\rightarrow \infty \) since it becomes possible to change the machine axes values with a diminishing effect on the pose. Therefore local to a singularity a machine may require relatively large speeds to attain a constant feed rate [7].
Consider the Hermle C600U with its kinematic chain. Certain orientations correspond to a singularity, these occur when \(\mathbf{{O}}_W=(0,0,1)^T\) (Table 1). This is due to the fact that when the cutter is oriented at \((0,0,1)^T\) it is possible to spin the C-axis and follow a circle in the XY plane centered on the C-axis of rotation without affecting the pose. This can be demonstrated with calculation of \(\det (\mathbf{{K}}^T\mathbf{{K}})\). For simplicity consider only the sub-matrix (\(\mathbf{{K}}_O\)) corresponding to orientation changes and rotary axes movements. Then
From (2)
Therefore
which gives
When \(\mathbf{{O}}_W=(0,0,1)^T\) then \(A=0\) and thus \(||\mathbf{{K}}_0^T\mathbf{{K}}_O||=||\mathbf{{K}}^T\mathbf{{K}}||=0\).
The case study presented in the following section analyses machine axes movements as \(\det (\mathbf{{K}}^T\mathbf{{K}})\rightarrow 0\) and the spectral norm of the Jacobian, \(||\mathbf{{J}}||\), increases.
4 Case Study One: Machine-Dependent Sources of Error
In this section an example of a tool path that produces machining errors (as defined in Sect. 2) with the Hermle C600U machine tool is examined. It is taken from a recent publication [7] wherein further details can be found.
The tool path motion is defined as
The cutter tip (\(\mathbf{{V}}_W\)) moves in a straight line whilst the cutter orientation (\(\mathbf{{O}}_W\)) rotates at a uniform speed. A zenith (orientation closest to singularity) is reached halfway along the tool path at angle of \(1^{\circ }\). A graphical representation of this tool path is given in Fig. 5 which also illustrates the orientation of the cutter as well as a simulation for the expected shape of the part after cutting.
Applying the inverse kinematics (see Sect. 3) machine axes values can be determined at each moment of the motion and hence the speed of axes movements can be calculated. Figure 6 illustrates the machine kinematics of the rotary axes. Halfway through the tool path it is noted that the C-axis is required to spin at a maximum speed of 199.0 rpm which has been calculated from (3) as
However the maximum speed of the rotary axes for the Hermle C600U is around 25 rpm, as stated by the machine tool manufacturers [10]. Therefore the CNC controller has to make a compromise on the desired tool path motion. From this analysis a surface defect is predicted between \(x=23\) mm and \(x=27\) mm.
After machining, the part was inspected with a 3D micro coordinate measurement machine and surface roughness measurement device [11]. The images taken (Fig. 7) illustrate the presence of a surface defect in the form of a discolouration of the material at the center of the part. Furthermore, a roughness profile measurement taken across the part indicates an increase in surface roughness local to the predicted affected region between \(x=23\) mm and \(x=27\) mm across the part (Fig. 8). The surface defect is thought to be melting of aluminium from dwelling as a consequence of singular behaviour.
The singular behaviour can be eliminated by changing the Jacobian, which is defined from the kinematic chain of the machine tool. This is achieved by either choosing a machine tool with a different kinematic chain or changing the existing kinematic chain. The latter can be achieved by reorientating the workpiece on the machine bed (with the use of a jig [7]) which effectively inserts an extra node into the kinematic diagram (Fig. 3). Consequently the tool path has to be reorientated.
Therefore a second tool path, based upon the previous tool path, is constructed through reorientation of \(10^{\circ }\) in the direction away from the singularity (so its zenith is \(11^{\circ }\) away). This tool path motion is defined as
A graphical representation of this tool path is given in Fig. 9 which also illustrates the orientation of the cutter as well as a simulation for the expected shape of the part after cutting. The corresponding machine kinematics of the rotary axes are given in Fig. 10.
Applying the reorientation has had the effect of reducing the C-axis speed from around 199.0 rpm to around 18.2 rpm, which has been calculated from (3) as
This value is below the C-axis maximum speed (Fig. 10) and is therefore not expected to cause the same issues as in the first tool path. The image in Fig. 11 confirms that the surface defect from the original tool path, explained as a consequence of singular behaviour, has been successfully removed through reorientation.
This case study illustrates that, by applying a shape-preserving transformation to reorientate the tool path, sources of error be can eliminated. Therefore the error cannot be attributed to the geometry of the tool path but rather the Jacobian and thus the machine tool kinematic chain. For this reason these errors are referred to as machine-dependent. In the next case study an example of a machining flaw that cannot be resolved with a manipulation of the kinematic chain is presented.
5 Case Study Two: Machine-Independent Sources of Error
Consider the task of machining a turbine blade as in Fig. 12 [3]. Machining flaws often occur on the turbine blade between the flatter sections and the more rounded edges (as highlighted in Fig. 13). To gain insight into what might be causing these flaws a simplified 2D tool path is considered. This turbine blade boundary consists of a flat section followed by a semi-circular tip leading back to another flat section. A machining strategy that preserves the angle between the surface normal and tangential direction (described in Sect. 2) is chosen for use with the Hermle C600U machine tool that maintains a uniform cutting feed rate.
The tool path is chosen to lie in the xy-plane. To ensure the cutter orientation remains in this plane the A-axis value is fixed at \(90^{\circ }\). The angle of orientation in the xy-plane corresponds to the angle of the C-axis. Along the flat sections the orientation of the cutter is preserved. This requires the rotary axes to be stationary. In the circular section the orientation of the cutter rotates through \(180^{\circ }\) at a uniform speed. This requires the C-rotary axis to be moving at a constant speed.
At the joins between flat sections and circular sections, the rotary axes have to transition from being stationary to moving at a constant speed in an instance. This corresponds to an infinite amount of acceleration/deceleration, which is physically impossible. The CNC controller must therefore make alterations to the tool path motion resulting in a different set of cutting conditions possibly leading to machining flaws.
Recall Eq. (3): \(\dot{\mathbf{{P}}}_M=\mathbf{{J}}\dot{\mathbf{{P}}}_W\). Along the flat sections the last three components of \(\dot{\mathbf{{P}}}_W\) are zero since these represent the change in orientation. Along circular sections the magnitude of the vector formed from the last three coordinates is a non-zero constant signifying the orientation speed. Thus there is a discontinuity in \(\dot{\mathbf{{P}}}_W\) where the sections join. Since the Jacobian is locally constant at the join between the flat and circular sections, the machine coordinates \(\dot{\mathbf{{P}}}_M\) must also experience a discontinuity and \(\ddot{\mathbf{{P}}}_M\) becomes unbounded.
An infinite acceleration of the machine axes therefore occurs independent of the choice for the kinematic chain of the machine tool. For this reason the error is referred to as machine-independent. Equation (3) can be used to distinguish between a machine dependent error and a machine-independent error. Machine-dependent errors relate to the Jacobian matrix whereas machine-independent errors relate to the \(\dot{\mathbf{{P}}}_W\) data. Thus a solution to eliminate machine-independent errors requires a reformulation of the pose data \(\mathbf{{P}}_W\). This can be achieved by either altering the machining strategy, such as no longer preserving the angles with normal and tangent, or by altering the CAD geometry.
6 Discussion: Geometric Contribution to Machine Errors
Case study one showed that singularity errors are machine-dependent whilst case study two showed that geometric discontinuities are machine-independent. However, not all errors are of these forms. A more realistic expectation is that errors are contributed from both components. Thus, given any machining error it is useful to quantify how much is caused by the machine-configuration and how much by the geometry to decide the best course of action. To ensure there are no machine-dependent errors, the Jacobian needs to be suitably bounded. To ensure there are no machine-independent errors, the \(\dot{\mathbf{{P}}}_W\) data needs to be suitably bounded. The pose data \(\mathbf{{P}}_W\) is inherited from the surface geometry of the CAD model. Therefore geometric properties of the surface affect the machining characteristics.
To avoid geometric discontinuities of the \(\mathbf{{P}}_W\) poses, the underlying CAD surface must satisfy certain continuity constraints. This section derives these constraints and then discusses approaches to bound the \(\dot{\mathbf{{P}}}_W\) data. To begin the following claim is proven: given a \(G^n\) surface, a curve can be obtained from the surface in which the normal vector is \(C^{n-1}\) smooth.
The unit normal can be defined in terms of first derivatives of curves on the surface. Given a non-degenerate parametrised surface, \(\mathbf{{S}}(u,v)\), the normal, \(\mathbf{{N}}(s)\), is derived from:
This corresponds to the normalised cross product of the tangent vectors for two iso-parametric curves at the point on the surface. Given any two curves that meet at the surface normal with non-parallel tangents a similar expression, replacing \(\mathbf{{S}}_u\) and \(\mathbf{{S}}_v\) with the new tangents, can be formed. Given that \(\mathbf{{S}}(u,v)\) is \(G^n\) it is possible to find two \(G^n\) curves [12] crossing at the surface normal with non-parallel tangents. The cross product of these tangents has \(C^{n-1}\) continuity since the cross product is a \(C^{\infty }\) operation and thus preserves the \(C^{n-1}\) continuity inherited from the tangents. Thus any \(G^n\) surface admits a \(G^n\) curve that delivers \(C^{n-1}\) continuity of the surface normal.
Given the surface, \(\mathbf{{S}}(u,v)\), a curve lying on it can be written as \(\mathbf{{C}}(t)=\mathbf{{S}}(u(t),v(t))\). A tool path motion that traces out such a curve has the following properties. The CC data is obtained from the arc-length parametrised curve of \(\mathbf{{C}}(t)\), referred to as CC(s). A local orientation frame, \(\mathbf{{R}}_F=\{F_x,F_y,F_z\}\), for this curve can be defined. The \(F_x\) direction is defined as \(\mathbf{{T}}(s)=CC'(s)\), the \(F_z\) direction is aligned to the surface normal and the unit bi-normal is the cross product of these \(\mathbf{{B}}(s)=\mathbf{{T}}(s)\wedge \mathbf{{N}}(s)\). This frame is commonly referred to as the Darboux frame [14].
The cutting tool can be orientated in any direction depending upon the desired machining strategy. The angles the cutter makes with the surface normal and tangent affect the chip pattern on the surface [4]. In five-axis machining efforts are usually made to preserve these angles to maintain a consistent surface finish. In this case, a fixed rotational offset of the Darboux frame can be defined as a unit vector, \(\mathbf{{O}}_O\), that forms the respective angles with the \(F_x\) and \(F_z\) axes. A rotation matrix with the axes of the reference frame as rows, \(\mathbf{{R}}_F=[\mathbf{{T}}, -\mathbf{{B}}, \mathbf{{N}}]\) can then be used to define the orientation vector in workpiece coordinates as:
The last step involves finding the CL values to complete the pose data \(\mathbf{{P}}_W\). The shape of the cutter can be interpreted as a surface of revolution and, when it touches a surface (that is it does not gouge), its normal intersects with the axis of revolution (see Fig. 2). The CL data, defined at the tip of the tool, can therefore be defined as a translation along the axis of revolution from the intersection point. This axis is defined from the orientation data and so it is possible to form the following equation:
The values of \(d_1\) and \(d_2\) depend upon the cutter geometry and the orientation of the cutter with respect to the Darboux frame (\(\mathbf{{O}}_O\)) which are assumed to be constant throughout the motion.
Therefore the components of the pose data comprise the underlying curve along with first derivatives of the surface. Algebra of limit arguments can thus be employed to show that \(G^n\) continuity of \(\mathbf{{C}}(t)\) delivers \(C^{n-1}\) continuity of the tool path motion \(\mathbf{{P}}_W(s)\). Note that not every curve that can be obtained from a \(G^n\) surface is \(G^n\) continuous and hence sufficient care must be taken when tracing out the CC curve on the surface. This highlights another potential cause of error in selection of the tool paths. For example, machining the simplified blade in Sect. 5 along the orthogonal direction could have avoided traversing the \(G^2\) discontinuity.
The occurrence of machine-independent errors, in the form of discontinuous axes movements, can thus be summarised with the following statement. Given a \(C^n\) smooth tool path from a \(G^n\) smooth surface the machine axes kinematics are \(C^{n-1}\) smooth. Note however that this assumes the Jacobian is \(C^{n-1}\) smooth and the angles between the surface normal and tangent direction are preserved.
In case study one the tool path was \(G^{\infty }\) smooth and there were no discontinuities in the \(\mathbf{{P}}_W\) data. In case study two the curve was \(G^1\) continuous and hence the machine coordinates were \(C^0\) continuous. However, there was a \(G^2\) discontinuity between flat and round sections and thus \(C^1\) continuity of the machine coordinates could not be achieved. Using a \(G^2\) curve and surface would ensure \(C^1\) continuous machine coordinates and thus finite acceleration bounds.
Bounding the kinematic properties of the machine axes is possible by bounding pose data and the Jacobian. Bounds on the machine axis speeds and accelerations [6] can be determined from Eq. (3) as
It is not enough for these bounds to exist but rather they must be less than a predetermined amount. Thus simply eliminating discontinuities does not eradicate machine-independent errors. Bounding the maximum velocity and acceleration to specific values is required to ensure no machine-independent errors.
In the scenario where the velocity and acceleration of machine axes are bounded but exceed kinematic constraints one possible solution to eliminate machine errors is to reduce the cutting feed rate. This approach is undesirable because other issues can occur with a reduced cutting feed, such as rubbing of the workpiece. However an important concern of productivity is that a lower feed rate necessitates longer machining times. In the ideal situation a tool path should admit as high as possible feed rate before exceeding kinematic constraints. This motivates further research into determining what type of geometry could preclude machine-independent errors whilst accommodating high feed rates.
7 Conclusions
Machining flaws can be predicted by analysing the desired tool path motion. A tool path motion can be converted into machine coordinates representing the values of the machine axes. If the expected speed or acceleration of an individual machine axis is too high then it differs from the desired motion. Consequently a loss of control in the cutting conditions is experienced potentially leading to machining flaws. This forms the basis of a diagnostic tool for analysing tool path motions. If the velocity or acceleration of the machines axes exceed predetermined bounds a machining error is said to occur. These should be eliminated if potential machining flaws are to be avoided.
The mathematical processes required to form the position and orientation (poses) of the cutter relative to the workpiece was presented in Sect. 2. Given knowledge of the machine tool structure, in terms of a kinematic chain, the relationship between poses and machine axis values was then given. Further insight into this relationship was gleaned from the Jacobian which represents how the coordinate systems change with respect to each other.
Two case studies were then presented. In the first case study the effect of a Jacobian with a large spectral norm, relating to a machining singularity, was investigated. It was shown that by changing machine tool structure, or more specifically the kinematic chain, these machining errors can be eliminated. For this reason machining singularities were identified as a machine-dependent error. In the second case of machining a turbine blade, a geometric discontinuity caused an infinite acceleration of one of the machine axes. It was shown that this occurs independent of the kinematic chain. Such errors were referred to as machine-independent errors.
Section six then discussed the differences between machine-dependent and machine-independent errors. These are characterised by two separate factors. The Jacobian characterises machine-dependent errors whereas the pose data characterises machine-independent errors. Thus the geometric contribution of the machinability of a surface comes from the pose data. Discontinuities in the pose data can be eliminated if the underlying surface possesses sufficient continuity conditions. It was argued that given a \(G^n\) continuous surface the time derivatives of the machine axes, up to the \((n-1)^{th}\) derivatives, are also continuous (assuming a sufficiently smooth Jacobian).
A surface with a high machinability must admit curves that yield desirable machine kinematic profiles. The derivatives of the poses represent the geometric contribution to the machine axes velocity and acceleration. These in turn are defined from the surface and constrained by the geometry that defines them. For example, a bumpy surface intuitively seems to be more difficult to machine than a smooth surface. What needs to be done next is obtain criteria for surface geometry under which it possesses good machinability properties. This will be the subject of future investigations.
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Acknowledgement
The research is supported by the EPSRC research council (EP/L010321/1 and EP/L006316/1). The authors also thank Delcam International PLC for supporting the research presented in this paper.
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Cripps, R.J., Cross, B., Mullineux, G., Hunt, M. (2017). Machinability of Surfaces via Motion Analysis. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_4
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