Digital Manufacturing – Part 1
Digital Manufacturing
Updated: 19/10/2026
Manufacturing systems
To give shape to a block of raw material (workpiece) there are three possibilities:
- Deform it: forming processes (motion axes: 1+)
- Cut it: material removal (or subtractive) processes (motion axes: 2.5+)
- Build on it: additive manufacturing processes (motion axes: 2.5+)
Manufacturing systems
- We skip over forming processes, where motion planning and control is trivial (with exceptions!)
- In subtractive and additive manufacturing there is a tool performing the process (a mill, a polarized wire, an extruding nozzle, …), whose motion is performed by synchronized action of two or more axes, typically in cartesian arrangement
- The motion (trajectory and speed) must be planned according to geometry and process specifics
Turning
Machining parameters to be calculated are spindle speed, feed rate, and spindle power:
\[ \begin{aligned} \text{Rotational speed:}\quad & N = \frac{v}{\pi D_0} \\ \text{Diameter:}\quad & D_f = D_0 - 2d \\ \text{Feed rate:}\quad & f = N f_r \\ \text{Machining time:}\quad & T_m = \frac{L}{f} = \frac{\pi D_0 L}{f_r v} \\ \text{Material Removal Rate:}\quad & MRR = v f_r d \end{aligned} \]
Drilling
Machining parameters to be calculated are spindle speed, feed rate, and spindle power:
\[ \begin{aligned} N &= \frac{v}{\pi D} \\ f &= N f_r \\ T_m &= \frac{t + A}{f} \\ A &= \frac{D}{2}\tan\left(\frac{\pi - \theta}{2}\right) \\ MRR &= \frac{\pi D^2 f}{4} \end{aligned} \]
Milling
Machining parameters to be calculated are spindle speed, feed rate, and spindle power. There are three types of milling:
Milling
\[ \begin{aligned} N &= \frac{v}{\pi D} \\ f &= N f_z z \\ MRR &= w d f \\ A &= \begin{cases} \sqrt{d(D-d)} & \text{side milling} \\ \sqrt{w(D-w)} & \text{face/end milling} \end{cases} \\ T_m &= \begin{cases} (L + A)/f & \text{side milling} \\ (L + 2A)/f & \text{face/end milling} \end{cases} \end{aligned} \]
Tool wear
Taylor’s law
\(VT^n=K \rightarrow \log(V) - n\log(T) = \log(K) \rightarrow \log(T) = \frac{1}{n}\log(V) - \frac{1}{n}\log(K)\)
Productivity optimization — Turning
- The optimal cutting speed \(V_{o,c}\) corresponds to maximum economic efficiency
- However, when there are penalties for delays it may be worth producing at \(V_{o,t}\), because it is the speed that maximizes productivity
- So, \(V\) shall be normally set to \(V_{o,c}\), increasing it up to (but not above) \(V_{o,t}\) if there are penalties for delays
Machine tools
Types of Lathes
- Parallel or engine lathe
- CNC lathe
- Horizontal spindle
- single spindle
- multiple spindle
- Vertical spindle
- Sliding headstock (fantina mobile)
- Horizontal spindle
Parallel (engine) lathe
Parallel (engine) lathe
- Programmable engine lathe
- Same architecture as an engine lathe
- Simple controller for programming simple motions (cylinder, cone, sphere, plane)
CNC lathe
- Direct-drive spindle
- Roller-turner (revolving) turret
- Counterstock (hydraulics)
- Inclined bed
- Lead screw
Moveable rest
Two-spindle, two turrets lathe
Multiple spindle lathe
Multiple spindle lathe
Vertical spindle lathe
Sliding headstock
Parameters
- Slewing (Brandeggio)
- Turning diameter
- Turning length
- X and Z travels (and possibly other axes)
- Spindle speed
- Rapid speeds
- Spindle torque and power
Milling machines
- Huge variety of architectures
- Common denominator: rotary tool
- From desktop size to 200 m\(^3\) footprint
- Some have rotary table and can do turning (hybrid mill-turn)
Types of Milling Machines
- CNC Milling Machines
- Horizontal spindle
- Vertical spindle
- Milling centers
- Universal
- Pallet changer
- Traveling column
- Architectures
- Cantilever
- Traveling column
- Gantry
- Parallel kinematics
Universal milling center
- Universal milling centers have a rotary table and a universal milling head that can be tilted and swiveled to machine any surface of the workpiece
- By rotating the axis at 45° from Z, the spindle axis can be changed from vertical to horizontal (universal)
Architectures
- Cantilever: simplest and cheapest
- Traveling column: suitable when one axis is much longer
- Gantry: much more stiff and stable, but more complex (twin axes)
- Parallel-K: much stiffer and faster, but complex to control and limited in travels
Vertical spindle
Pros:
- Easy fixturing
- Reduced moveable mass
Cons:
- Chip disposal is more difficult
- Not suitable for machining dies/cavities
Horizontal spindle
Pros:
- Better chip/liquids disposal
- Large travels
Cons:
- More difficult fixturing
- More moveable mass (slower)
- more difficult to fixture
Pallet changer
- Stocky WPs
- High productivity
- can be integrated in automated lines
Traveling column
Suitable for:
- slender WP
- stocky WP but with two alternating fixturing zones (higher productivity)
Masked time
Any operation that can be performed in parallel to machining is said to happen in masked time. Its cost is absorbed in the machining cost
Other operations happen in exposed time, which is added to the machining cost
Milling centers
Parameters
- Axes configuration
- Axes travels
- Spindle speed
- Rapid speed
- Spindle torque and power
Automation
It is mainly a matter of synchronizing motion
Current level: A(4)
- CNC machine tools
- Convert a machining description in machining movements
- Run machining commands with judgement (A4)
Machine tool elements
Drivers
Requirements:
- Fast response (acceleration)
- Quick stabilization (minimum number of oscillations)
- No overshoot (or risk of damages)
Feedback
Errors
Linear axis control
Encoder gain: pulses/rev — Axis gain: screw pitch × encoder gain (mm/pulse)
Linear guide-rail
- Lube port
- end wiper seal
- end cap
- runner block body
- rolling element
- side seal
- guide rail
PID controllers
| Parameter | Rise time | Overshoot | Settling time | Steady-state error | Stability |
|---|---|---|---|---|---|
| Kp | Decrease | Increase | Small change | Decrease | Degrade |
| Ki | Decrease | Increase | Increase | Decrease significantly | Degrade |
| Kd | Minor decrease | Minor decrease | Minor decrease | No effect | Improve if Kd small |
Actual control schemes
PLC: Programmable Logic Controller
PLC is a controller that implements logic relationships between inputs and outputs. It provides a range of interfaces and registers:
- Analog and digital I/O (> 2048)
- Relay (1000)
- Memories (200)
- Timers (200)
- Counters (200)
Note
Can be either a separate hardware, or a software process running on the same computer that runs the CNC
PLC: Ladder diagram
Note
- On the left side there are contactors (inputs), on the right side there are coils (outputs).
- parallel branches represent OR logic, series branches represent AND logic; contactors can be normally open or normally closed (NOT).
- The program is read from left to right, top to bottom. The program is executed in a cyclic scan: all inputs are read, the logic is evaluated, and the outputs are updated.
PLC: Function Block Diagram
Note
- Each block implements a function, which can be a logic function, a timer, a counter, or even a more complicated one as a PID controller
- Appeals to control engineers, being quite similar to Simulink and other block-diagram-based control design software
- Complex programs tend to become quite messy, so FBD is often used in combination with other languages and limited to implement specific functions related to control and signal processing
PLC: Sequential Function Chart
Note
- SFC is a graphical language that implements a finite state machine
- Each block is a state, each line a transition with its activation condition
- It is very intuitive for process engineers, and it is often used in combination with other languages to implement the overall logic of a process, while the other languages are used to implement the details of each state
Coordinate systems
- Axes references combine to give machine origin
- Axes references are used to reset encoders (homing)
- Machine controller works in machine coordinates
- Workpiece origin is set up by the operator
- Registration point is the reference to measure tool length and diameter
- Support point is the reference to mount the tool on the spindle (when mounted, reference = support)
| Reference | Symbol |
|---|---|
| Axis reference |  |
| Machine origin |  |
| Workpiece origin |  |
| Registration point |  |
| Support point |  |
Tool reference
- Support point:
- Abutment plane (lathes)
- Spindle nose (mills)
- Registration point:
- on the tool, the point that matches the SP when mounted
- To be measured:
- Length and offset (lathes)
- Length, diameter, and toric radius (mills)
Coordinate transformations
Any coordinate \(P_{wp}\) given in the workpiece reference frame \(O_{wp}\) is converted in machine coordinates \(P_m\), including tool length offset
\[ P_m = T_{wp \to m} P_{wp}; \qquad T_{wp \to m} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & d_x \\ r_{21} & r_{22} & r_{23} & d_y \\ r_{31} & r_{32} & r_{33} & d_z + l_t \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
\(d_{x-z}\) are coordinates of \(O_{wp}\) in \(O_m\)
Coordinate transformations
Remember that the roto-translation matrix is:
\[ T = \begin{bmatrix} \cos\alpha\cos\beta & \cos\alpha\sin\beta\sin\gamma - \sin\alpha\cos\beta & \cos\alpha\sin\beta\cos\gamma + \sin\alpha\sin\gamma & d_x \\ \sin\alpha\cos\beta & \sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\beta & \sin\alpha\sin\beta\cos\gamma - \cos\alpha\sin\gamma & d_y \\ -\sin\beta & \cos\beta\sin\gamma & \cos\beta\cos\gamma & d_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
The order of transformations is:
- Roll by \(\gamma\)
- Pitch by \(\beta\)
- Yaw by \(\alpha\)
- Translate by \((d_x, d_y, d_z)\)
Measuring positions: touch probe
Reference positions (as \(O_{wp}\)can be measure in machine thanks to a touch probe:
- a special non-rotating tool with a deflecting stylus and spherical tip that can be used to touch the workpiece and measure its position in machine coordinates
- The 3 contact points provide a kinematic mount are also electric switches
- When a switch opens, the current coordinate plus the sphere radius in the direction of motion gives the touch point coordinate
Tool reference (presetting)
Tool geometry can be measured in machine (exposed time) or by a tool presetter (masked time). The latter is preferred because it is faster and more accurate, and it allows to measure the tool geometry without having to mount it on the machine.
- On-board presetting uses a fixed stylus/switched contact point in known position reachable by the tool
- Tool presetting systems have a dummy spindle mount, a camera and a shadow projector
Automation
According to the Yardstick of Automation:
- most CNC machines are at Level A4
- Level A5 requires post-machining automated inspection and feedback to the CNC controller and to the part-program
- a validated process model can be required to implement A5, which is not necessarily available for most machining processes
- empirical models (machine learning) can be used to implement A5, but they require a lot of data and are not always reliable
Machining cycle
The general sequence for machining a part is:
The section within the red box is the machining cycle, i.e. the sequence of operations that can be cyclically repeated for each part
Part-program structure
The structure of a part program generally follows a common structure:
- setups
- roughing section: removes the bulk of the material, usually with a large tool and high feedrate, leaving a small allowance for finishing
- finishing section: removes the remaining material with a small tool and low feedrate (final geometry)
- reset of the machine to a safe position and end of program
Toolpaths
- Roughing and finishing may require different tools for different geometries
- Roughing typically follows simple, stepped trajectories
- Roughing allowance (excess material) is usually in the range of
- 0.5–1 mm for turning
- 0.2–0.5 mm for milling
Punch tape
- G-Code was developed during 1950s
- Computers only had punch tapes as input devices
- Language had to be concise
Example: turning
Example: turning — roughing
N10 G00 X61.0 Z7.0
N11 G01 Z-94.5 F1000.0
N12 G01 G91 X5.0 F2000.0
N13 G00 Z7.0
N14 G00 X51.0
N15 G01 Z-88 F1000.0
N16 G01 G91 X5.0 F2000.0
N17 G00 Z7.0N18 G00 X41.0 Z7.0
N19 G01 Z-59.4 F1000.0
N20 G01 G91 X5.0 F2000.0
N21 G00 Z7.0
N22 G00 X32.0
N23 G01 Z-57.6 F1000.0
N24 G01 G91 X5.0 F2000.0
N25 G00 Z7.0Example: turning — finishing
N31 G00 X70.0 Z0.0
N32 G00 X35.0
N33 G01 X0.0 F1000.0 S2000.0
N34 G00 Z5.0
N35 G00 X30.0
N36 G01 Z-20.0
N37 G01 X25.0 Z-25N38 G01 Z-52.0
N39 G02 X45.0 Z-62.0 I10.0 K0.0
N40 G01 Z-85.0
N41 G01 G91 X20.0 Z-15.0
N42 G01 G90 Z-150.0
N43 G00 X70.0
N44 G00 Z20.0
...
N98 M30Example: milling
- WP origin is usually placed in \(y,y\) plane on a geometry feature where most of the dimensions are referenced to, e.g. the bottom-left corner of a rectangular pocket
- The \(z\) coordinate is set to the topmost surface on the finished part:
- the very first roughing operation will remove the excess material and reach the finished surface, which will be used as reference for all subsequent operations
- any subsequent operation is carried out at a negative \(z\) coordinate, which is the depth of cut below the finished surface
Example: milling
G-code language
%es1 G71 *
N10 G30 G17 X+0 Y+0 Z-20*
N20 G31 X+150 Y+150 Z+0*
N30 G247 Q339=+1*
N40 T1 G17 S5000 F4000*
N50 G00 G90 X+14 Y-15 Z+10 G40 M8*
N60 G00 Z-10 G40*
N70 G01 Y+95 G41 F4000*
N80 G01 X+36 Y+140*
N90 G25 R10*
N100 G01 X+105*
N110 G24 R4*
N120 G01 G91 Y-55*
N130 G25 R14*
N140 G01 G91 X+35*
N150 G25 R5*
N160 G01 G90 X+105 Y+14*
N170 G24 R8*
N180 G01 X-15*
N190 G00 Z+10 G40*
N200 G38 M30*
N99999999 %es1 G71 *
Program check
G-code language
%es1 G71 *
N10 G30 G17 X+0 Y+0 Z-20*
N20 G31 X+150 Y+150 Z+0*
N30 G247 Q339=+1*
N40 T1 G17 S5000 F4000*
N50 G00 G90 X+14 Y-15 Z+10 G40 M8*
N60 G00 Z-10 G40*
N70 G01 Y+95 G41 F4000*
N80 G01 X+36 Y+140*
N90 G25 R10*
N100 G01 X+105*
N110 G24 R4*
N120 G01 G91 Y-55*
N130 G25 R14*
N140 G01 G91 X+35*
N150 G25 R5*
N160 G01 G90 X+105 Y+14*
N170 G24 R8*
N180 G01 X-15*
N190 G00 Z+10 G40*
N200 G38 M30*
N99999999 %es1 G71 *
CNC
Algorithms for Computer Numerical Control — machining complex shapes
Profiling
Profiling is where motion algorithms are implemented. For every block:
- preserve block length: \(\displaystyle\int_{t_a}^{t_b} f(\tau)\,\mathrm{d}\tau = l\)
- limit acceleration to the maximum allowed by the machine
- limit jerks, to minimize vibrations and mechanical stress
Profiling
Discrete time-base (clock steps of \(t_q\)) and feedrate quantization. Block durations must be scaled to a multiple of \(t_q\) yet maintaining the same block length. The actual feedrate is reduced accordingly.
\[ \Delta t_1 = \frac{f_m - f_s}{a}, \quad \Delta t_2 = \frac{f_m - f_e}{d}, \quad \Delta t_m = \frac{l}{f_m} - \Delta t_1 \frac{f_m + f_s}{2 f_m} - \Delta t_2 \frac{f_e + f_m}{2 f_m} \]
\[ \Delta t^* = \mathrm{int}\left(\frac{\Delta t}{t_q} + 1\right) t_q = \Delta t_1 + \Delta t_m^* + \Delta t_2, \qquad \Delta t_m^* = \Delta t_m + \Delta t^* - \Delta t \]
Profiling
Discrete time-base (clock steps of \(t_q\)) and feedrate quantization. Block durations must be scaled to a multiple of \(t_q\) yet maintaining the same block length. The actual feedrate is reduced accordingly.
\[ \int_{t_s}^{t_e} f(\tau)\,\mathrm{d}\tau = \frac{\Delta t_1}{2}(f_m^* + f_s) + \Delta t_m^* f_m^* + \frac{\Delta t_2}{2}(f_m^* + f_e) = l \]
\[ f_m^* = \frac{2l - f_s \Delta t_1 - f_e \Delta t_2}{\Delta t_1 + \Delta t_2 + 2\Delta t_m^*}, \qquad a^* = \frac{f_m^* - f_s}{\Delta t_1}, \qquad d^* = \frac{f_e - f_m^*}{\Delta t_2} \]
Profiling
In short segments there’s no time enough to reach the nominal feedrate: \(\Delta t_m \le 0\)
\[ \int_{t_s}^{t_e} f(\tau)\,\mathrm{d}\tau = \frac{a\Delta t_1^2 + d\Delta t_2^2}{2}; \quad a\Delta t_1 = d\Delta t_2 \]
\[ \frac{\Delta t_1^2 (a + a^2/d)}{2} = l \]
\[ \begin{aligned} \Delta t_1 &= \sqrt{\frac{2l}{a + a^2/d}} \\ \Delta t_2 &= \frac{a}{d}\Delta t_1 \\ \Delta t_2^* &= \Delta t^* - \Delta t_1 \end{aligned} \qquad f_m^*(\Delta t_1 + \Delta t_2^*) = 2l \qquad \begin{aligned} a^* &= \frac{f_m^* - f_s}{\Delta t_1} \\ d^* &= \frac{f_e - f_m^*}{\Delta t_2^*} \end{aligned} \]
Motion interpolation
- The speed profile is evaluated and integrated at each time tick: \(\lambda(t) = 1/L\int_0^t f(\tau)\,d\tau\)
- The axes set-points are calculated with the parametric equation of the path
- For a linear motion:
\[ \begin{aligned} X(t) &= X_0 + \Delta X \lambda(t) \\ Y(t) &= Y_0 + \Delta Y \lambda(t) \\ Z(t) &= Z_0 + \Delta Z \lambda(t) \end{aligned} \]
Motion interpolation
For arc interpolation:
\[ \begin{aligned} X(t) &= X_c + R\cos(\vartheta_0 + \Delta\vartheta \lambda(t)) \\ Y(t) &= Y_c + R\sin(\vartheta_0 + \Delta\vartheta \lambda(t)) \\ Z(t) &= Z_0 + \Delta Z \lambda(t) \end{aligned} \]
Warning: feedrate may be reduced
\[ f_{m,a} = \min\left(f_m, \sqrt{aR}\right) \]
Motion interpolation
Two alternative G-code syntax for arcs:
- arch center:
G[02|03] X[\(X_f\)] Y[\(Y_f\)] I[\(I\)] J[\(J\)] - radius:
G[02|03] X[\(X_f\)] Y[\(Y_f\)] R[\(\pm R\)](radius sign discriminates which arc to take)
\[ \begin{aligned} \vartheta_0 &= \cos^{-1} I/R \\ \vartheta_f &= \cos^{-1} (X_f - X_c)/R \\ X_c &= X_0 + I \\ Y_c &= Y_0 + J \\ R &= \sqrt{I^2 + J^2} \\ R^2 &= (X_f - X_c)^2 + (Y_f - Y_c)^2 \end{aligned} \]
Jerk
Jerk-limited profiling:
- corners in feedrate imply discontinuous acceleration
- discontinuous acceleration mean infinite jerk
- infinite jerk produces vibrations and mechanical stress
- for limited jerk profiling each segment is split into 7 ranges (vs. 3)
Look-ahead feedrate
Step 1: calculate velocities:
- \(v_{i,i} = v_{f,i-1}\), or 0 for first sg.
- \(v_{m,i} = f_i\) for G01
- \(v_m = \min(f_i, \sqrt{aR})\) for G02/G03
- \(v_{f,i} = \dfrac{v_{m,i} + v_{m,i+1}}{2}\cos\alpha_i\)
- \(v_{f,i} = 0\) for last segment or if \(v_{m,i} = 0\)
Forward step: accelerations
Long segments, \(v_i < v_m\):
A. \(v_f > v_m\), \(s_0 < s_1 < s_2 < s_f\)
B. \(v_f < v_m\), \(s_0 < s_1 < s_f < s_2\)
Forward step: accelerations
Long segments, \(v_i > v_m\):
A. \(v_f > v_m\), \(s_1 < s_0 < s_2 < s_f\)
B. \(v_f < v_m\), \(s_1 < s_0 < s_f < s_2\)
Forward step: accelerations
Short segments:
- \(s_0 < s_f < s_1\)
- \(v_f\) and \(s_2\) are not relevant
Forward step: accelerations
Short segments:
- If \(v_i\) is large enough, fallback to long segment cases
Backward step: decelerations
- Same approach as forward step
- For each block, select the path going from \((s_0, v_i)\) to \((s_f, v_f)\)
9 Possible cases
Errors in machine tools
Machine tools are precision systems — laser interferometry
Profiling errors
Look-ahead can alter trajectory, especially at corners:
- More evident at high speed
- When a sharp corner is needed, use curls or disable look-ahead
Feedrate-related errors
Positioning errors
Squareness errors
Laser interferometer
Laser interferometer
Linearity measurement
- This is the simplest measurement
- The laser beam is reflected by a mirror mounted on the axis and the interferometer measures the distance travelled by the axis
- It is a way to calibrate and to verify the positioning accuracy of the axis encoder
Angle measurement
- An angle interferometer only changes the optical paths difference upon rotation
- Rotation can happen either on the splitter or in the retro-reflector
- The action plane can be selected to measure the pitch or the yaw angles
Straightness measurement
A Wollaston prism is a beam-splitting prism that splits a beam of light into two beams with orthogonal polarizations. The two beams are diverging at a small angle (typically 15–30 arcmin). The prism is used to measure the straightness of an axis by measuring the lateral displacement of the reflected beam.
Straightness measurement
- By aligning the Wollaston prism action plane with XY and with XZ planes it is possible to measure the straightness of the X axis in Y and Z directions
Squareness measurement
- By combining lateral displacement measurement with an optical square it is possible to measure the squareness between two axes.
- The optical square is a reflecting prism that reflects the laser beam at 90° its incoming direction
Calibration
Laser interferometer: three-axis machine tool installation
High precision machine tools are equipped with a laser interferometer to measure the actual position of the axes.
- one laser head (emitter only)
- one receiver per axis
- beam splitters direct 1/3 of the total source power to each axis
Note
Laser beam can be affected by refraction index changes and by air turbulence, so the machine is usually equipped with a temperature and pressure sensor to correct the measured distance and the laser is usually enclosed in a protective tube to avoid air turbulence.
