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Mathematics Analysis and Approaches (AA) Extended Essay Exemplar:… | RevisionDojo
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IB Mathematics Analysis and Approaches (AA) SL Extended Essay Example
Comment détermine-t-on la position, l’orientation et le mouvement de l’élément final d’un bras articulé à 6 degrés de liberté en connaissant les angles articulaires des différentes composantes du bras? (How do we determine the position, orientation and movement of the final element of a 6-degree-of-freedom articulated arm knowing the joint angles of the different components of the arm?)SL
A
Official IB Result
28/34
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Criteria A: Focus and Method
6/6
0
3
6
Criteria Strands
A.1Topic communication and explanation
Excellent
A.2Research question clarity and focus
Excellent
A.3Research methodology
Excellent
Criteria Feedback
Research question is clearly stated, narrowly focused on 6‐DoF manipulator kinematics and revisited throughout the report
Topic is communicated accurately with a coherent introduction, notation section, and sustained focus
Methodology is comprehensive—combining analytical derivations, numerical validation and a bespoke C# simulator—and sources are relevant
Minor digressions into peripheral robotics examples (ASIMO/NAO) slightly expand scope
A brief definition of DH parameters or alternative conventions would sharpen methodological context
1.1·Suggestion
Page 8• Click to view
A brief definition of Denavit–Hartenberg parameters would strengthen the modelling chapter by situating the homogeneous transform approach within standard robotics conventions.
1.2·Suggestion
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The choice of Cartesian coordinates is justified briefly; consider explaining why alternatives (e.g. DH frames) were not used to highlight informed methodology.
1.3·Suggestion
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The wrist joints (θ4,θ5,θ6) are introduced without explicit transform matrices here; include their homogeneous transforms for completeness.
1.4·Suggestion
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List of Euler conventions is thorough; highlight the chosen Y–Z–Y'' sequence earlier when defining orientation method to improve focus and avoid reader confusion.
Criteria B: Knowledge and Understanding
6/6
0
3
6
Criteria Strands
B.1Subject knowledge application
Excellent
B.2Use of terminology and concepts
Excellent
Criteria Feedback
Derivations of 2‐, 3‐, and 6‐DoF forward kinematics demonstrate strong command of linear algebra and robotics concepts
Consistently relevant application of subject knowledge to the research question
Accurate and consistent use of specialized terminology and notation, as evidenced by the comprehensive notation table
Occasional sign slips (e.g. unconventional frame orientations) and duplicated symbols introduce minor ambiguity
Small typographical lapses in subscript notation interrupt the otherwise consistent use of terms
2.1·Weakness
Page 12• Click to view
In equations (3.13)–(3.15) the sign convention on y2=−L2cosϕ2 is unconventional; clarify the frame orientation or correct the sign slip to avoid confusion.
2.2·Suggestion
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The general transform in (3.23) is clear, but the transition to the 3×3 block lacks explanation; briefly derive the bottom row [0 0 1] for completeness.
2.3·Suggestion
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When introducing three-joint control in 3.2.3, the rationale for adding a torsional joint is sound, but referencing a standard spherical wrist configuration would reinforce subject understanding.
2.4·Weakness
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The expansion to 6 DoF in (3.33) uses Tn(θn,Tn) but duplicates the T symbol; revise notation to Tn(θn,Ln) to avoid ambiguity.
2.5·Suggestion
Page 20• Click to view
Equations (4.10) list X1=0;X2=0 without derivation; briefly justify these initial conditions for better clarity of the projection steps.
Criteria C: Critical Thinking
12/12
0
6
12
Criteria Strands
C.1Research quality
Moderate
C.2Analysis depth
Moderate
C.3Discussion and evaluation
Moderate
Criteria Feedback
Research is drawn from authoritative texts and integrated into an original simulator, maintaining consistent relevance to the research question
Multi‐step derivations and numeric examples demonstrate clear logical progression and effective support for conclusions
Well‐structured argument with recognition of limitations and future directions
Limited novel theoretical contribution beyond standard textbook derivations
Absence of quantitative error analysis or robustness testing of the methods
Critical evaluation of alternative formalisms (e.g. DH vs. homogeneous transforms) is only briefly mentioned
3.1·Suggestion
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Validation description is clear, but the projection approach in section 4.1 lacks error quantification; consider adding a numerical error analysis to strengthen evaluation.
3.2·Suggestion
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Equation (4.13) repeats sin(θ2+θ3) for X5 but ignores θ4; the text notes this but formal derivation would improve analytical rigour.
3.3·Suggestion
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The derivation of X3=L2sinθ2cosθ1 (4.11) is correct but would benefit from a small diagram indicating the two successive projections.
3.4·Suggestion
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The added correction term L5sinθ5cosθ4cos(θ2+θ3)cosθ1 (4.15) is well identified, but include its derivation via geometric relationships for completeness.
3.5·Suggestion
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Derivation of Euler angles via R22=cosβ (4.31) is correct; note potential gimbal-lock when R22=±1 and discuss handling of edge cases.
3.6·Suggestion
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Inverse kinematics introduction defines the concept well; consider referencing how redundancy in 6 DoF arms complicates general solutions, even though out of scope.
3.7·Suggestion
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The numeric example (5.22) rounds α to four decimals then degrees; specify rounding policy consistently to maintain numerical transparency.
3.8·Suggestion
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Conclusion effectively summarizes the work and future directions; to enhance critical evaluation, reflect on specific methodological limitations encountered.
Criteria D: Presentation
4/4
0
2
4
Criteria Strands
D.1Structure appropriateness
Excellent
D.2Layout and formatting
Good
Criteria Feedback
Logical chapter structure and clear progression support comprehension
Key presentation elements—title page, pagination, word count, typeset mathematics, labelled figures—are all in place
Consistent formatting of code annex and notation table enhances professionalism
Encoding artifacts (e.g. HTML entities, accented character glitches) detract slightly from polish
Inconsistent indentation in the table of contents and minor margin misalignments appear
Some notation (inverse‐tan) is not fully standardized across the document
4.1·Weakness
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The section header shows encoding artifacts (“math {…}”); standardize accented characters and remove OCR anomalies for professional polish.
4.2·Suggestion
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Table of contents presents a logical chapter flow, but indentation of sub-sections is inconsistent; align entries and page numbers uniformly.
4.3·Weakness
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Notation table effectively defines symbols, but mixing HTML entities (“&”) in matrix entries interrupts readability; use pure LaTeX formatting consistently.
4.4·Weakness
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Equation (3.20) uses HTML entities in the rotation matrix; render cosθ and sinθ in pure LaTeX to improve presentation consistency.
4.5·Suggestion
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Table 4.4 clearly shows correction signs, but add units (e.g.\
4.6·Suggestion
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The combined expression for X6 in (4.17) is compact but hard to parse; recommend spacing and line breaks in LaTeX to enhance readability.
4.7·Weakness
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Matrix multiplication in (4.29) is correctly executed, but the final equals sign is disconnected; ensure alignment of equals and matrix blocks for clarity.
4.8·Suggestion
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Simulation screenshot lacks axis labels and units on graphs; include labels to connect numerical output with the kinematic model effectively.
4.9·Weakness
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Using \texttt{atan} in (4.40) deviates from previous use of tan−1; standardize inverse-tan notation throughout for consistency.
